3:49

I've made the observation that they're both one more than a multiple of eight,

Â and then I sign off by stating that this is, this in fact proves the theorem.

Â So I've got a beginning, a middle, and an end, so it succeeds communicatively.

Â I've given reasons for each things. Notice I didn't just square these things.

Â I didn't simply square them, I said I was squaring them, so that the reader's not

Â left wondering what I'm doing. Now, to red it might be a little bit

Â [INAUDIBLE] why I started in this way and that's basically experienced

Â mathematicians can fairly regularly moderately come up with this kind of

Â thing. If you're a beginner it can take you

Â longer. Okay, and then when I've dome that

Â calculation I make a specific conclusion, and this is a conclusion.

Â A conclusion to the reasoning, not to the whole proof, because the conclusion of

Â the whole proof is the observation that this proves the theorem.

Â This is the drum roll, this is where the cymbals crash, this is where we take a

Â bow. We've done it, we're out of there.

Â Okay, so the answer is yeah, it's valid. And we just checked that it's valid by

Â going through all of these arguments, okay?

Â Let's move on to number three. Well, this proof is in fact valid, okay?

Â Let's just make sure that it works on all levels.

Â we're arguing by contradiction, okay? So, this is the beginning and this is the

Â beginning step, suppose the conclusion's false.

Â Then there'll be a natural number n for which a is not true, so there'll be a

Â least one. Now about the first condition, in our

Â case, at A one, we know that the m can't be equal to one, because A does at one,

Â so m is bigger than one. So m equals n plus one for some n.

Â Since n is less than m, we know that n holds because n is at least one of which

Â it doesn't. Then by the second condition, A n plus

Â one holds, i.e., A m. That's a contradiction, and that proves

Â the result. the only thing to notice is that here the

Â n has been used in, in a quantified form, so this is a variable that's been

Â quantified, right? Okay, in this case, what we're saying is

Â if the, if the, if it, if this fails, then there'll be one of these n's for

Â which it fails. And at this stage, the n is sort of a

Â variable. But at these points, when we put the m

Â in, m is a specific number. We don't know what its values is.

Â In fact, we're going to show that no such m exists.

Â But on the basis of a false assumption, there will be a, a place where it fails.

Â And m is going to be a specific number, so the m here is specific within the

Â proof. Okay, when I get down to here, the n is

Â also specific within the proof. Because it's equal to m minus one.

Â So in these cases, once I'm down here, I've got a specific m and a specific n.

Â Up here, it's just a general n, okay? So everything here is specific.

Â 7:19

Now, in fact, there never was such an m, and hence there never was such an n.

Â Because the conclusion as a result is valid.

Â But within the text of this argument, these guys are specific.

Â Here, there's a, there's a, this is a variable that's quantified, alright?

Â In any case this is proof this is true. On number four, we have to use the course

Â rubric again. first of all, let's see what, what the

Â theorem says. It says that if we take the Fibonacci

Â numbers. And we square them and we take the first

Â n of them and square them another. Then the result is equal to the nth

Â Fibbionaci number, the last one in this sequence here, multiplied by the one

Â after that. now this is actually, this is true, by

Â the way. This is true, it's a valid theorem.

Â It's one of many identities about Fibonacci numbers that show that they're

Â connected in In what at first, a very surprising ways.

Â they're very typical in, that they're almost always proved by induction.

Â And when you look at the induction proof, you realize that it's really that the

Â identity, which at first, seems surprising.

Â Is actually just a disguised version of the definition of the Fibonacci numbers.

Â What's behind the, all of these interesting identities.

Â Is the fact that the the n plus second Fibonacci number is the sum of the nth

Â Fibonacci number and the n plus first Fibonacci number.

Â So they all go back to the fact that it's defined in this iteratively additive way,

Â okay? well, let's just see if this one, see,

Â see how this one goes, okay? first of all, we'll look at logical

Â correctness. worrying about things like reasons and so

Â forth later. Let's check that the, the first case is

Â true. F1 is equal to one, so F1 squared, and in

Â the case, n equals one, there is no sum it's just F1 squared.

Â So the left hand side is just one, on the right hand side is F1, which is one.

Â And of course the second Fibonacci number is also one.

Â Because the Fibonacci sequence begins with a pair of ones.

Â So we have two ones on the right. So, this is logically correct.

Â then there's the, the induction step. Let's just check the algebra here.

Â here it's just the taking taking a sum up to n plus one, and pulling out the second

Â one, the last one, the last term in the series.

Â So we've got the sum of n plus 1 is the sum of the first n together with the last

Â one. we'll look at the issue of, of reasons in

Â a minute. this will be the induction hypothesis,

Â and nicely stated as, as a reason. that this sum equals Fn plus one, this is

Â just Fn plus one squared I carried through.

Â take out Fn plus one as a common factor okay.

Â let's see, this is definition of Fn plus two, is the sum of that one plus that

Â one. And then we've, we've got the other

Â identity n plus 1. Okay, so, so all of the logical steps are

Â correct. This is a valid induction proof.

Â So I'm going to give four marks for that. is it clear?

Â Yes, I think this is clear. again, there's going to be some issues of

Â reason as to explain things, but everything is clear.

Â It's well laid out, it's easy to follow steps, even when I had to figure out what

Â the author meant. I'm going to, we'll talk in a minute as

Â to whether, whether I should have had to. I, it, it was, it was easy to follow in

Â that sense. So it, so it was clear.

Â there's an opening, it's well opened. It's a proof by induction, there's a

Â standard method. And, and they, they, the good way to

Â start. The correct, well, almost to start to

Â prove using a standard method is to state what the method is.

Â So I want to get full marks for that. conclusion when the conclusion is stated

Â I'll, I'll look at whether it's properly stated in a minute.

Â But it certainly it, it, stated when the proof is complete And it's been laid out

Â that it's going to be an induction proof. So, I think we're going to get four for

Â that reasons, a couple of quibbles. I think the order should have said,

Â separate out fine term. this is good, stating the, the use of the

Â induction hypothesis and an induction proof is, is always, is critical.

Â I, I, I think that's this is such an important step.

Â by algebra, yeah, you could say something like check out a common factor but the,

Â this is, this level of mathematics. When we're doing proofs in number theory

Â as I mentioned with the last problem set. we, we can assume that people can, can

Â spot things like taking out common factors.

Â this I think is important. and this, this this is critical, the fact

Â that we're using the, the definition of the, of the Fibonacci numbers.

Â the, these kinds of identities, as I mentioned a moment ago, these kind of

Â identities actually only hold because of the way the Fibonacci sequence's identity

Â is, is defined. which establishes the identity for n plus

Â one. That's good that's a local conclusion,

Â which is good. the proof is complete, here the person

Â should have said, by induction, or by the principle of induction.

Â let's put out in four principle of induction, because there is a powerful

Â fact about what the, the natural numbers that's been used here.

Â Okay, so, what I'm going to do for reasons, I'm going to give two, I think.

Â Because I can't give four. you know, I, if, if this one arguably I,

Â I, I would tend to lean on having this in especially since it's an opening step in

Â the proof. But if that was the only thing that was

Â missing in this context, I might not have even deducted any marks at all.

Â this one however, I, I have to deduct at least one, and I think, I think I've

Â really deducted, I think really I've deducted the two for this one.

Â It was, it was a bit of a judgement call. you know, I have to allow for the fact

Â that the author of a proof maybe made a slightly different judgement call.

Â you, you have to try, judge how well you think a persons putting down a proof.

Â and, and you can't really say, I always do it this way, therefore you always

Â should do it this way. Because people have different, they come

Â on different sides of these issues. we're really looking to grade this as an

Â overall thing. You know, one of the problems with using

Â a rubric is we're trying to take something that's holistic.

Â And is basically an overall judgement call and reduce it to number of

Â parameters. This is not how professionals go ,they

Â look overall and say ,this is a good proof and then assign a number ,but now

Â space for years of expertise. Pulling us apart this way is a good for

Â beginners because it allows you to focus on one only individual things, but a

Â professional looks at all these things in, in one.

Â And you have to sort of balance things out.

Â and that's why we're going to all these videos, to try and give you some

Â indication of how a professional, and in this case the professional is me.

Â How we go about it and how we, tacitly, and when we're doing this, this part of

Â our every day work, when we're grading work.

Â whether, whether we're grading student's work or we're evaluating proofs of other

Â mathematicians. This is all part of the tacit process of

Â grading. and, and in writing out a proof, I've

Â just, tried to isolate the things that I implicitly and automatically look at in

Â grading proofs. as indeed does, does any professor,

Â professional mathematician when they're grading a proof, okay?

Â and so, and what I'm really saying is, this is.

Â Not really perfect in terms of giving reasons.

Â arguably this one is more important than that one, for example.

Â You know, it's overall, however, I'm going to get four.

Â because these are sort of niggling. and the reason they're niggling is the

Â person has laid out the fact that this induction, has made it clear that they're

Â assuming it for n, have proved it for n plus 1.

Â they've definitely stated the use of the induction hypothesis.

Â as I said, this one would have been nice, I would have liked to see it, but this

Â author. Presumably decided it really was, was,

Â was patently obvious what's going on. this I think is an important one, because

Â that's critical to the proof. this is not something that's typical.

Â The farther you can pull this in depends on the, where the Fibonacci sequence is

Â defined, and this is the only part in the proof where we make use of that fact.

Â So you really should Should mention this one.

Â So this ones important, and this one's important.

Â And the reason is until we've got to the last line, all we've really done is we've

Â shown that the thing is true for an equals one, two.

Â Actually, further equals one, use the further and F2 was what it was.

Â So if there were few, they were put through for the first case, or observed

Â in the first case. And then we've shown that if it holds

Â stage n, it follows, and it holds stage n plus one.

Â So we've proved two simple facts. One, the fact about the first one, well,

Â it's the first two for that matter. And secondly, we've proved an implication

Â from n to n plus one. The conclusion is that this holds for

Â infinitely many number, for all of the actual numbers.

Â So somehow proving, two statements, one simple observation and one implication,

Â has proved that something is true for the infinitude of all natural numbers.

Â Now admittedly, induction has a natural it's always an obvious thing, this sort

Â of a, self evident truth to the principle of induction, you know.

Â You can think of it in terms of, of knocking rows of dominoes over or

Â something. So But the fact that the, that the, that

Â the principal of introduction, or the part, the method of induction has got a

Â sort of intuitive obviousness to it. Shouldn't obscure the fact that this is

Â actually a deep result. Making a conclusion about an infinite set

Â is non trivial, you know? The, the Hilbert Hotel tells us that

Â infinity's a very, paradoxical domain. We've got to be very careful.

Â So, this actually holds. We can make the conclusion that it's true

Â for all n. Because of the principle of mathematical

Â induction. This, in other words, this is a big deal.

Â This is a big, big deal. And when big deals are involved, you

Â should mention them. You know, if there's a big guy in the

Â room. it's polite, if not, [LAUGH] a matter of

Â self preservation to observe that fact and make it clear.

Â So you really do need to state the principle of induction here to state that

Â it's been used, or at least to say, by induction.

Â I, I, you know, if I was feeling, if it wasn't for the fact that the rest of the

Â proof was laid out so nicely. I might well have just deducted more

Â here. But as it is given everything else was

Â laid out so well and given that the proof is nice and elegant.

Â And this is a slick proof, there's almost no superfluous lines in.

Â And I think overall simply deducting two marks is about right.

Â So I've got 22 out of 24 for this one. and I feel reasonably good about this.

Â you know, the, couple of a small points, [COUGH] well, it's one small point, one

Â moderate big points, one really huge point, I think.

Â I think, this is being generous. But I think this is proof deserves

Â generosity. overall, I'm happy with that.

Â And, and this is really how it, it splits up.

Â Okay, let's go ahead and look at number five.

Â And number five is another of these Fibonacci sequence results.

Â This one says that if we take the first n Fibonacci numbers, add them together, the

Â result is the the next but one. Fibonacci number, we skip over Fn plus

Â one, we go to Fn plus two. Okay, so, okay, so as is typical for

Â these results it is proved by induction. So see how the proof goes.

Â For n equals one, the left-hand side is F1, that's why you get if n equals one,

Â there's no sum, it's just F1 itself. And the right-hand side is uh-oh, oh

Â dear. If n equals one, the right-hand side

Â would be F3 and F3 equals two, and one does not equal two.

Â So this isn't even true, it's not valid for n equals one, which means the

Â theorem's not valid. Oh, good grief, this is such an obvious

Â mistake. It's the kind of mistake that anyone

Â could make. Doesn't really reflect on their ability

Â as a mathematician. It's just a human error.

Â So common this kind of thing, and yet this is mathematics.

Â Ultimately in mathematics things are right or wrong.

Â I mean, you know, if, if this mathematics being used by an engineer to build a

Â bridge, and the bridge falls down and people get killed.

Â You know, that, that engineer could be held liable.

Â i am just defining or so you know at the end of the day we call that thing goes to

Â false ,i am have to 0 for logical correctness before, we are go further ,i

Â mean it is just a plan to evolve false result you could make the result correct

Â ,you could make first result correct by subtracting one And it fact it turns out

Â as, and I'll come back to this, that if you put a minus 1 in here, then the

Â identity is true for all n. So there is a theorem here, and the get,

Â we get at the theorem by noticing what went wrong with this proof.

Â Incidentally, this is very typical in mathematics.

Â often in mathematics, the statement of the theorem when it's proved isn't the

Â one that the author originally tried to, to do.

Â Very often in mathematics we, we make a conjecture.

Â We try to prove it. the proof has gone wrong and so by

Â analyzing the proof we've thought of go back and change the statement.

Â So it's often the case that statements of theorems actually come after the proof.

Â 21:57

Not many proofs in mathematics began as proofs of something else that failed, but

Â then the, the statement has changed to what's been proved.

Â So it's not always the case that the mathematicians sort of formulated

Â theorems and then proved them. They often formulate a theorem, develop a

Â proof, find out the proof is wrong, go back and restate the theorem so the proof

Â works for that restatement. Okay, that's just the way mathematics

Â advances, it's, it's part of the process of, of getting you knowledge.

Â Okay, [COUGH], well we're going to have to come back and, and, and sort of look

Â at how the thing works as a proof but, but it, it isn't a proof as it stands.

Â Okay, well let's go through the mechanics of it and see if, if all of the other

Â steps are okay. So, assume the identity holds for n, then

Â well what's going on here? This is a case of separate, I'll, I'll,

Â I'll give grades for, for these ones later, but let's just mention that what's

Â going on here is separate out the last term.

Â Okay, let's see, that's the reason here. Sum up to n plus one is the sum up to n

Â together with the final term. this is induction hypothesis.

Â That guy, equals that. Incidentally and I'm jumping ahead to

Â when I corrected this, at this point there should be a negative one in here,

Â and if we do that it carries through. So this, this is the proof as we're going

Â through it will work validly. Providing that we stare at the theorem

Â correctly. Okay, so I'm not going to give double

Â jeopardy deduction for this. We've already knocked a ton off for that.

Â okay, so the rest of the thing is, is actually, the logic is correct, in of

Â itself. Expect for this glaring mistake at step

Â one. But that, that topples the whole

Â interphase. So I'm going to give four for clarity

Â because it's absolutely clear, well laid out.

Â it's proof by induction, it states it, stated the method at the beginning.

Â So four for the opening, the conclusion is certainly stated, I'm going to give

Â four marks for that. What's the problem I have with this is

Â that, it's not mentioned that this is an induction proof.

Â And as I elaborated on a great length with the previous question, was question

Â four. the fact that something is, you know the

Â fact of an infinite three/g, and infinite three statement.

Â The statement about something which we've all end.

Â The fact that that follows from a couple of little facts like an observation and a

Â simple implication that's a big deal. and that takes us into the realm of the

Â infinite. And what takes us into the realm of the

Â infinite is the fact that we have this thing called The Principle of

Â Mathematical Induction. Which is difficult to prove if you try to

Â prove it, or you have to assume it is, it is an axiom or some kind of principle.

Â So this is not a trivial thing. It's intuitively clear, I know.

Â But it's not it's, by no means easy to prove it.

Â So you know, their, we're pulling on something powerful here.

Â We should state that we're pulling on something powerful.

Â Okay, no reasons, I mean I was addressing those as I went through.

Â this ones missing, that's in, that's good.

Â You know, the, the, the, the fact that the, the definition of the, I mean it,

Â thi, this res-, this result holds, I mean, when I modify it to make it true.

Â It holds by virtue of the way that Fibonacci sequence is defined.

Â So you should stipulate the fact that you're using that.

Â this is good, using the induction hypothesis.

Â I've got something good here, I've got something good here.

Â that's a bit of a problem. That should've been in.

Â this one, I, you know, again, this is this is a judgement call, I would like to

Â see that there, but that's just me. All of the things being equal, I would of

Â ignored that, but all of the things I'm seeing, of course, this person's already

Â missed this out. So, got a missing reason here, missing

Â reason here, but a couple of good ones here.

Â so I'm going to give two, okay. I think it's about right, overall

Â valuation? The thing is false, I mean, I can't

Â possibly give four overall for a theorem that's plain false, you know.

Â Much as I, I'm sympathetic for the fact that it was a simple slip, the most I can

Â give is two. And I think I'm being generous there,

Â quite frankly because it is a false result.

Â On the other hand the, the, the course is focusing on mathematical thinking and

Â mathematical communication, the ability to formulate and present a proof, and

Â there's lots of aspects to that. It's not just about whether things are

Â right or wrong. You know high school mathematics is, is

Â largely focused on, K through 12, mathematics is largely focused on things

Â being right or wrong. we've, we've moved beyond that now.

Â Right and wrong is still an important factor which is why I give zero to this

Â part. But there's other things we're looking

Â at. So I will be giving 16 for this one,

Â okay? Well, that's said eight max lost out of

Â 24, so this pair lost a third of the max for this.

Â So one could say that this is generous for false proof for false result.

Â On the other hand there's a lot of good stuff here.

Â It was simply a silly mistake right here at the beginning.

Â Okay, that was very unfortunate. Okay, life's like that at times, let's go

Â on and look at number six. Well, this is another one about the

Â Fibonacci sequence. what does it say?

Â It says that the n Fibonacci number is at least equal to the number three over two,

Â to the power n minus two. Okay, let's see how this person does

Â this. we have, oh, okay.

Â Interesting way to start, making a statement about F1.

Â which is equal to one, true, I mean, I, I wouldn't look at logical correctness.

Â I'll just follow it through, and see what this person's doing.

Â and two over three is equal to three over two to the negative one.

Â Okay, well I guess that's showing that it's true for F equals F1.

Â so maybe this is an induction proof, although there's no way of knowing just

Â by looking at it, right? well this is interesting.

Â The person now goes on and, you know, you usual with an induction proof you prove

Â it for n equals 1 and then you stop. This person now proving it for F 2.

Â well, let's see what they're doing. So they're saying F2 equals one, which is

Â does. And one is indeed three over two to the

Â power zero, which is true, because anything to the power zero equals one.

Â So the inequality is valid for n equals one, two, alright, absolutely correct.

Â Now assume the inequality holds for n, where n greater or equal to two, so, we

Â assume, but we're not told, that this is going to be an induction proof.

Â This person just jumped in, so this is, this is already not a, not a proof.

Â This isn't telling a story. It's, it's, it's, it's presenting us with

Â a, with a who done it or a what done it. Or what are they doing?

Â so this is going to be a mystery, where n's greater than or equal to two.

Â Okay, then, let's see. Fn plus one equals Fn plus Fn minus one.

Â True, that's a definition of the Fibonacci sequence.

Â Although there's no explanation of that fact.

Â that's greater than or equal to, well, ha.

Â Where's this coming from? well looking ahead to the fact that this

Â is the, the person's is almost certainly doing an induction proof, but hasn't

Â written it, written it down. What should of been said here, was for

Â wholes the in, inequality wholes up to and including it that, I think, is what's

Â is what's meant here. Because we're assuming, we're using it

Â for two cases here. We're using it for the two previous

Â cases, and we're saying that is greater than or equal to that one, that's greater

Â than or equal to that one. and then then we're taking out a common

Â factor. this is interesting, there's almost no

Â reasons given for anything else. And certainly the person makes the

Â obvious statement that this is bialgebra. Which indeed it is, taking out 3 over 2

Â to the nc is a common factor. And one of them now is a digit of one

Â there. then 3 over 2 plus one equals 5 over 2.

Â Now we're spelling everything out in goal we have mathematical detail.

Â 5 over 2 is 10 over 4, why are you doing that?

Â Well, because because then you can put a 9 over 4 here and make it smaller.

Â And then nine over four is just 3 over 2 squared.

Â Which gets you everything back to 3 over 2.

Â Which establishes the inequality for n plus one.

Â Okay well, this person can certainly manipulate fractions and also really

Â impressive. Because you know a large percentage of

Â the world's population has trouble dealing with fractions and inverting

Â fractions and things. So, so this person has a lot of

Â procedural skill with fractions, but is it a proof?

Â Heavens no, I mean this, this so much missing here.

Â I'm going to give four for logical correctness, because the manipulations

Â and the logic and everything was, was, was fine.

Â as a professional mathematician, able to figure out what's going on.

Â I could recon I am, and I could recognize fairly early, fairly early on that this

Â was an induction proof. But no thanks to the person writing it

Â down, and just because I've got a lot of experience.

Â never the less I'm going to give four for the, for the logical correctness.

Â I'm just going to get four for the clarity, because it was sufficiently well

Â lit out. That one side realized that there's an

Â induction proof going on here. I was able to follow the steps.

Â so I didn't have to solve, bury myself in a, in a marage of details.

Â To find out what was going on it was clear.

Â Opening, well there wasn't one, there absolutely wasn't one.

Â This person just straight in, and did so very obscurely, because even if looking

Â at the first case indicates that we're doing induction.

Â Why look at the second case when we almost never do that in induction.

Â And then we had to make a, a modification to, to even make sense of what was going

Â on. So, no opening, what about the

Â conclusion? Well no, there isn't a conclusion.

Â what the person presumably should have said, or maybe meant to but didn't.

Â Hence, by induction, we could simply say the theorem is proved, that's fine.

Â That's absolutely fine. It bears fruit, this is alright.

Â but it's missing. There's no conclusion stated, and

Â there'll be a zero here. Reasons, there aren't any.

Â There aren't any reasons. simply no reasons.

Â I mean, this one doesn't count. I'm not going to give credit for that.

Â That's just a, here, we should have said something like well, by, by the

Â induction, well actually, there's two things.

Â By the definition, of the Fibonacci sequence is the first equality, okay?

Â And the second one, is the induction hypothesis.

Â 33:49

This is by the definition of the Fibonacci sequence.

Â you know, it should be said. I mean, this is so self evident.

Â and either way it's written, because this is literally the definition.

Â You know, you could maybe, let this person have have some, some leeway, if

Â everything else was good. But nothing else is good, so I'm not

Â going to, I'm, I'm not inclined to give credit generously.

Â When the so lack of, there's so much lack of reasons give.

Â This is the induction hypothesis applied twice, okay?

Â Once for that one and once for that one, and then the rest is algebra.

Â Okay overall, the only you know I do, you know, if, if I simply leave it at that

Â And say that this is just terrible. Because it is terrible as a proof.

Â The person gets eight out 24, and yet, this is some, this is quite sophisticated

Â by most people's standards. I'm going to give two for the overall.

Â And I'm going to give some sort of compensatory credits in here.

Â For the fact that this person's done some really intricate, manipulations.

Â there's some intricacy in here, this is not trivial but it still just gives 10

Â out of 24, I mean this is a, a low mark. Now in terms of a grade for mathematical

Â dexterity that would be low, this person clearly has considerable mathematical

Â dexterity. But proving things and communicating

Â things in mathematics is much more than dexterity.

Â it's about telling a story, it's giving reasons.

Â It's establishing why something is true, you know?

Â I'm prepared to believe that this person convinced themselves that this is true,

Â and they understood why it was true. But what they haven't done is express

Â that fact properly. this doesn't even come close to being a

Â proof, as a proof, this one sucks. I mean, this, this is not a proof at all,

Â okay? no mention of reasons.

Â we've still got this mystery of why did the person.

Â Why did this person do the first two cases?

Â Well that was actually an obser, a good observation.

Â mostly with induction proofs, you have to prove.

Â Just, you, you still have to prove the first case.

Â But this one, you have to prove two of them.

Â Because if you look at the induction step, it uses it twice.

Â You have to use it as the two previous cases.

Â So in this case, the induction step depends on having it twice.

Â You've got Fn greater than or equal to that.

Â And you've got Fn minus one greater than or equal to that.

Â So we've got two instances of the induction hy-, the induction hypothesis

Â here. Well, it would, the induction hypothesis

Â has to be pulled in twice. We use two inequalities.

Â So you have to prove the first two cases. And you also have to, when you, when

Â you're making the assumption. You're not just assuming it for n you're

Â assuming it n and n minus one. Okay, you're assuming it for n and n

Â minus one. and, and the simple way to say that is

Â you assuming it up to and including n. So there's a lot in terms of missing

Â reasons here. the fact that the proof, the fact that

Â this person does what he or she does Indicates to me that they almost

Â certainly understand these issues. You know, simply observing you have to do

Â it for two cases, is significant. So I've got a lot of sympathy with this

Â person. They've obviously thought deeply.

Â I'm convinced, actually, that they know what they're doing.

Â And they know why they're doing it, otherwise, they wouldn't have done this.

Â