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[The following is summary of Wildberger’s video lecture. You will find that he is a supremely talented teacher. Any mistakes are my own, as I am not a mathematician. Bracketed comments are my own and are not to be trusted.]
Norman J. Wildberger
Course Series
Math Foundations
Math Foundations B (80-149)
Real Numbers and Limits
111
“Real numbers and Cauchy sequences of rationals (I)”
111.1-111.8
(00.08-10.07)
“Defining Cauchy Sequences]”
Brief summary:
(111.1) We will examine and criticize the thinking behind Cauchy sequences. (111.2) Cauchy was a brilliant and prolific mathematician who founded complex analysis, among other things. (111.3) We look now at a Cauchy sequence. We will examine the formal definition given in analysis texts. (Wildberger claims that this definition does not quite logically work.) [The basic idea seems to be the following. What is a Cauchy sequence? It is a series of increasing rational numbers (that progressively tend toward some determinate value), such that no matter how small of a value we choose, there will always be a place in that series after which the differences between any two such values will be less than that arbitrarily small, chosen number. In other words, they converge upon a limit.]
‘A sequence S1, S2, S3 of say, rational numbers is a Cauchy sequence precisely when for all ε greater than zero there is an N, a natural number, with the property [N] or such that if little n or little m are bigger than or equal to N, then the difference between Sn minus Sm is less than ε. In words, what it means is that a Cauchy sequence is a sequence that has the property that after a certain point, all the elements in the sequence are close to each other. And a little bit more precisely, it means that no matter what level of tolerance ε you choose, as long as it is a positive level of tolerance, that there is some point that you can get to, denoted by this capital N, so that for all elements of the sequence past that point, the difference between any two of them is within ε.’
(01.42)
This is a classical definition with serious problems; nonetheless, it is the foundation of the idea that real numbers are actually Cauchy sequences of rationals. (111.4) The reason that the notion of a Cauchy sequence is logically flawed is a similar reason as that for the limit of an arbitrary sequence being a number A. What is problematic so far is that we have not yet defined what a sequence is. (111.5) We wonder, why is this definition important? What role does it play in constructing the real numbers, supposedly? There is a important key fact or theorem. [It says, basically, that a Cauchy sequence is one where a series of rational numbers tend toward a limit.]
‘If S1, S2, S3 is a sequence of rational numbers, with the limit of Sn = A a rational number, then this sequence is a Cauchy sequence. So if we have a sequence of rational numbers which actually does have a limit, in the classical sense of the limit that we defined in the last few videos, then that sequence is a Cauchy sequence. So sequences with limits are Cauchy sequences.’
(04.08)
(111.6) There is a standard proof of this theorem (although Wildberger does not think it is adequate.) [The basic point here seems to be that we can take that ε value (for which the differences between successive values after some point remain smaller than ε) and divide that ε in two, placing one half above the limit value and one half below it, creating a band range of values surrounding equally the limit value. The proof shows that once the terms enter into that band, they stay within it.
‘The idea is that once the terms of the sequence, whatever a sequence is, are within ε of 2 of this limit A, then they are within ε of each other.’
(4.52)
This can be explained with a diagram.
‘Here we have a sequence, S1, S2, S3, S4, S5, S6, etc.’ [pointing to the the first six blocks in series.] ‘So the values of the sequence are here on the y-axis. So S1 is whatever this value is,’ [pointing to the corresponding place on the y-axis for the first block] ‘S2 is this value, S3, is this value. And here is the value A,’ [pointing to the green line], ‘which we are assuming this limit is. So this sequence goes to A. Now, because it goes to A, we know that if we say choose some band, and let us choose the band to be ε over 2,’ [pointing to the band between dashed red lines], ‘where ε is that number given to us in the Cauchy sequence definition. So if we are given ε, then first of all we calculate ε over 2, and then we find an N’ [points to the N], ‘so that past that point, the sequence will be within ε over 2 of A, so between A plus ε over 2 and A minus ε over 2. So in this case here, the sequence is bouncing around, but let’s say that N = 5 and beyond, the sequence then manages to stay within this band around the value A. Well in that case, the difference between the values of the sequence of any two values beyond this N will necessarily be less than ε, because the total width of this band from top to the bottom is ε over 2 plus ε over 2, which is epsilon. So the argument is that any two of these sequence elements past this point N will be within at most ε of each other.’
(5.15-07.02)
(111.7) This can be said in a more mathematically precise way. [It seems to say that no matter how small the ε interval, there will be a place along the sequence after which the values will fall in the ε region either above or below the limit value toward which the sequence is tending.]
‘Given an ε greater than 0, find N, a natural number, so that if n is bigger than or equal to N, then Sn minus A, in absolute value, is less than ε over 2. We can do that, this is possible, since we are assuming that the limit of the sequence Sn is some value A.’
(07.06)
(111.8) This theorem can be formulated in terms of a triangle inequality. [It seems to say that for any two places past the natural number (selected because after it the numbers stay within the arbitrarily small selected ε value that surrounds the limit value), the difference between those values will be less then their total distances away from the limit, and thus they will remain within that ε value, which is equal to the top half of the band added to the bottom half of the band. This is the standard proof that Cauchy sequences are ones with a limit, and this implies that the sequence of values, pairwise, get closer and closer to that limit value.]
‘So if we believe this, then if n and m are bigger than or equal to N, in other words, beyond this value 5 in this example, then if we look at the difference between Sn and Sm, so Sn minus Sm in absolute value, this is less than or equal to the absolute value of Sn minus A plus the absolute value of A minus Sm. This is a triangle inequality, so a basic fact about inequalities. So that if you have two numbers Sn and Sm, say on the number line, then that separation is less than or equal to, well if you pick any number A whatsoever anywhere and you can look at the separation between Sn and A and between A and Sm, the sum of those two has to be bigger than or equal to the separation between Sn and Sm. So that’s the triangle inequality, basic fact about inequalities. And now we are assuming that n and m are past this point N. So from what we have assumed up here, we know that Sn minus A is going to be less ε over 2, this is less than ε over 2. And similarly A minus Sm, which in absolute value is the same as Sm minus A, is also less than ε over 2. And so the sum of these 2 is less than ε over 2 plus ε over 2, which is ε, showing that, yes, once little n and little m are bigger than this capital N that we have found, then the difference between any of two of these elements is less than ε. So that is the standard proof that shows that sequences with a limit are Cauchy sequences. That is an important fact to remember. If a sequence does have a limit, then it is Cauchy sequence. So if a sequence has a limit, it implies something about the sequence itself independent of the limit. The existence of that limit implies that the sequence elements themselves will be, pairwise, getting closer and closer to each other.
(07.46-10.07)
[Introduction to the Topic]
[Brief Bio of Cauchy]
[The Formal Definition of Cauchy Sequences]
[A Problem with the Definition]
[A Theorem about Cauchy Sequences Tending Toward a Limit]
[The Standard Proof of That Theorem]
[A More Mathematically Precise Formulation of the Theorem]
[The Theorem in Terms of a Triangle Inequality]
Summary
[Mostly identical to the brief summary above.]
[Introduction to the Topic]
[We will examine and criticize the thinking behind Cauchy sequences.]
(00.08-00.34)
[Wildberger says that
In today’s video we are going to look at Cauchy sequences, and we are going to start investigating why real numbers as “equivalence classes” of Cauchy sequences is really a very flawed idea.
]
[Brief Bio of Cauchy]
[Cauchy was a brilliant and prolific mathematician who founded complex analysis, among other things.
]
(00.35-01.07)
[ditto]
[The Formal Definition of Cauchy Sequences]
[We look now at a Cauchy sequence. We will examine the formal definition given in analysis texts. (Wildberger claims that this definition does not quite logically work.) [The basic idea seems to be the following. What is a Cauchy sequence? It is a series of increasing rational numbers (that progressively tend toward some determinate value), such that no matter how small of a value we choose, there will always be a place in that series after which the differences between any two such values will be less than that arbitrarily small, chosen number. In other words, they converge upon a limit.]
‘A sequence S1, S2, S3 of say, rational numbers is a Cauchy sequence precisely when for all ε greater than zero there is an N, a natural number, with the property [N] or such that if little n or little m are bigger than or equal to N, then the difference between Sn minus Sm is less than ε. In words, what it means is that a Cauchy sequence is a sequence that has the property that after a certain point, all the elements in the sequence are close to each other. And a little bit more precisely, it means that no matter what level of tolerance ε you choose, as long as it is a positive level of tolerance, that there is some point that you can get to, denoted by this capital N, so that for all elements of the sequence past that point, the difference between any two of them is within ε.’
(01.42)
This is a classical definition with serious problems; nonetheless, it is the foundation of the idea that real numbers are actually Cauchy sequences of rationals.]
(01.08-03.08)
[ditto]
[A Problem with the Definition]
[The reason that the notion of a Cauchy sequence is logically flawed is a similar reason as that for the limit of an arbitrary sequence being a number A. What is problematic so far is that we have not yet defined what a sequence is.
]
(03.09-03.52)
[ditto]
[A Theorem about Cauchy Sequences Tending Toward a Limit]
[We wonder, why is this definition important? What role does it play in constructing the real numbers, supposedly? There is a important key fact or theorem. [It says, basically, that a Cauchy sequence is one where a series of rational numbers tend toward a limit.]
‘If S1, S2, S3 is a sequence of rational numbers, with the limit of Sn = A a rational number, then this sequence is a Cauchy sequence. So if we have a sequence of rational numbers which actually does have a limit, in the classical sense of the limit that we defined in the last few videos, then that sequence is a Cauchy sequence. So sequences with limits are Cauchy sequences.’
(04.08)]
(03.53-04.40)
[ditto]
[The Standard Proof of That Theorem]
[There is a standard proof of this theorem (although Wildberger does not think it is adequate.) [The basic point here seems to be that we can take that ε value (for which the differences between successive values after some point remain smaller than ε) and divide that ε in two, placing one half above the limit value and one half below it, creating a band range of values surrounding equally the limit value. The proof shows that once the terms enter into that band, they stay within it.]
‘The idea is that once the terms of the sequence, whatever a sequence is, are within ε of 2 of this limit A, then they are within ε of each other.’
(4.52)
This can be explained with a diagram.
‘Here we have a sequence, S1, S2, S3, S4, S5, S6, etc.’ [pointing to the the first six blocks in series.] ‘So the values of the sequence are here on the y-axis. So S1 is whatever this value is,’ [pointing to the corresponding place on the y-axis for the first block] ‘S2 is this value, S3, is this value. And here is the value A,’ [pointing to the green line], ‘which we are assuming this limit is. So this sequence goes to A. Now, because it goes to A, we know that if we say choose some band, and let us choose the band to be ε over 2,’ [pointing to the band between dashed red lines], ‘where ε is that number given to us in the Cauchy sequence definition. So if we are given ε, then first of all we calculate ε over 2, and then we find an N’ [points to the N], ‘so that past that point, the sequence will be within ε over 2 of A, so between A plus ε over 2 and A minus ε over 2. So in this case here, the sequence is bouncing around, but let’s say that N = 5 and beyond, the sequence then manages to stay within this band around the value A. Well in that case, the difference between the values of the sequence of any two values beyond this N will necessarily be less than ε, because the total width of this band from top to the bottom is ε over 2 plus ε over 2, which is epsilon. So the argument is that any two of these sequence elements past this point N will be within at most ε of each other.’
(5.15-07.02)]
(04.41-07.02)
[ditto]
[A More Mathematically Precise Formulation of the Theorem]
[This can be said in a more mathematically precise way. [It seems to say that no matter how small the ε interval, there will be a place along the sequence after which the values will fall in the ε region either above or below the limit value toward which the sequence is tending.]
‘Given an ε greater than 0, find N, a natural number, so that if n is bigger than or equal to N, then Sn minus A, in absolute value, is less than ε over 2. We can do that, this is possible, since we are assuming that the limit of the sequence Sn is some value A.’
(07.06)]
(07.03-07.45)
[ditto]
[The Theorem in Terms of a Triangle Inequality]
[This theorem can be formulated in terms of a triangle inequality. [It seems to say that for any two places past the natural number (selected because after it the numbers stay within the arbitrarily small selected ε value that surrounds the limit value), the difference between those values will be less then their total distances away from the limit, and thus they will remain within that ε value, which is equal to the top half of the band added to the bottom half of the band. This is the standard proof that Cauchy sequences are ones with a limit, and this implies that the sequence of values, pairwise, get closer and closer to that limit value.]
‘So if we believe this, then if n and m are bigger than or equal to N, in other words, beyond this value 5 in this example, then if we look at the difference between Sn and Sm, so Sn minus Sm in absolute value, this is less than or equal to the absolute value of Sn minus A plus the absolute value of A minus Sm. This is a triangle inequality, so a basic fact about inequalities. So that if you have two numbers Sn and Sm, say on the number line, then that separation is less than or equal to, well if you pick any number A whatsoever anywhere and you can look at the separation between Sn and A and between A and Sm, the sum of those two has to be bigger than or equal to the separation between Sn and Sm. So that’s the triangle inequality, basic fact about inequalities. And now we are assuming that n and m are past this point N. So from what we have assumed up here, we know that Sn minus A is going to be less ε over 2, this is less than ε over 2. And similarly A minus Sm, which in absolute value is the same as Sm minus A, is also less than ε over 2. And so the sum of these 2 is less than ε over 2 plus ε over 2, which is ε, showing that, yes, once little n and little m are bigger than this capital N that we have found, then the difference between any of two of these elements is less than ε. So that is the standard proof that shows that sequences with a limit are Cauchy sequences. That is an important fact to remember. If a sequence does have a limit, then it is Cauchy sequence. So if a sequence has a limit, it implies something about the sequence itself independent of the limit. The existence of that limit implies that the sequence elements themselves will be, pairwise, getting closer and closer to each other.
(07.46-10.07)]
(07.46-10.07)
[ditto]
Wildberger, Norman J. (2014). “Real numbers and Cauchy sequences of rationals (I) | Real numbers and limits Math Foundations 111.” Part 111 of the course series: Math Foundations. Video.
Youtube page for this video.:
Course Youtube Playlist:
Math Foundations A (1-79)
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Math Foundations B (80-149)
https://www.youtube.com/playlist?list=PLIljB45xT85DpiADQOPth56AVC48SrPLc
Math Foundations C (150 - )
https://www.youtube.com/playlist?list=PLIljB45xT85AYIeGfDQwHM8i6PQEDnnTI
Norman J. Wildberger, youtube channel:
[njwildberger]
Insights into Mathematics
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