What exactly is a limit?? | Real numbers and limits Math Foundations 106 | N J Wildberger

In this video we aim to give a precise and simpler definition for what it means to say that: a rational polynumber on-sequence p(n) has a limit A, for some rational number A. Our definition is both much simpler and more logical than the usual epsilon -delta definition found in calculus texts. What is required is that we need to find two natural numbers: k called the scale, and m called the start that allow us to bound in a pretty simple way the difference between p(n) and A.

The epsilon-delta definition of a limit is usually considered a high point of logical rigour. Not so. It is also considered too logically involving to be taken seriously as a pedagogical pillar for most undergrads. Hence students may be told about the definition, but are not required to seriously understand it, or be able to use it--unless they are prospective maths majors.

There is a subtle ambiguity in the definition: given an epsilon we are supposed to demonstrate there is a delta (with certain properties) but how are we to do this, since an potential infinity of epsilons are involved? In practice what is required is a correspondence (function/relation etc) between epsilon and delta but the nature of this required correspondence is not clear. We return to our familiar conundrum of using the work``function'' without a proper definition of it.

The key point that makes our simpler more intuitive notion of limit of a sequence work is that we are dealing with very particular and clearly defined on-sequences: those generated by a rationl polynumber. A good example of the benefits of being careful rather than casual when dealing with the foundations of analysis!

Video Content:
00:00 Introduction
2:49 Definition of a limit
5:25 Definition of the limit of a sequence"
19:02 Problems with "limit of a sequence"
21:01 Rational polynumber on-sequences
23:28 Some obvious limits
25:34 Definition of limit (new!) with k and m
29:57 Constant sequence
31:36 An example and an exercise

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[jaanuskiipli]

I personally don't have a problem accepting epsilon-delta paradigm and I'm completly happy accepting formulas for calculation of delta (or N) from epsilon. And as you said, we can work in the realm of rationals and hence define real numbers as limits.

[rbrowne]

Always fascinating to hear Norman's ideas on these matters. He is a tornado ripping lots of the old obscure ideas to shreds. He then brings in his clearer and better defined ideas. These are much more satisfying.

[ako-math]

Thank you for great videos Mr. @njwildberger, what about sequences that converge slower than 1/n ? Say, can we talk about convergence in a sense that -k/n <= (p(n) - A)^2 <= k/n ?

[ull]

Your Math Foundation series are a super series Thank you.

[johnsterling]

wow loved this video. i kept trying to figure out exactly what a limit is but struggle to understand it so i searched "what exactly is a limit"

[jeremiekoenig]

Hi Prof Wildberger, thanks for another great video. I would like to point out there are indeed computer programs which are able to check proofs of this form. In fact, the constructive real numbers have been formalized at least in the Coq proof assistant (see for instance "Certified Exact Transcendental Real Number Computation in Coq", Russel O'Connor). Of course they don't have decidable equality, which as you point out in an earlier video is probably theoretically impossible.

[AlgebricDiddle]

Ok! Thank you! I'll wait for the next videos! :)

[Igdrazil]

What is certain is that in MOST case, except few academic ones built up to work smoothly for students, it can be extremely difficult to FIND EXPLICITELY the FORMULA linking epsilon and eta, and possibly explicitely IMPOSSIBLE!

[alvinlepik]

Not at all rigorous, but it might come to that at some point. Putting out some thoughts.

"Sequence" is not bestowed. It's defined to be a function that sets every natural number in correspondence with an element of a fixed non-empty set. A function is defined, in algebra, to be a relation (another correctly defined notion).
Loosely speaking, in a relation we have a correspondence between two non-empty sets X, Y. If, however, any x in X is in correspondence with exactly one y in Y, then we call such relation a function.

Real numbers are defined to be elements of a complete ordered field (introduced, actually, in analysis in view of the problems you pointed out), thus avoiding the circular reasoning defining real numbers via sequences and sequences, somehow, via real numbers. It is also proven that complete ordered fields exist and there is exactly one such field up to isomorphism.

We don't even need real numbers to define a sequence, in general. (In a topological space, for instance)

Ok, furthermore - absolute values: another way of saying -eps<f(x)-A<eps, no mystery here.
The requirement of such definition is very straightforward. Given an arbitrary positive value, our goal is to find an index from which point on, said criterion is satisfied.


My question now becomes: why do we set these constraints of studying only rational/polynomial expressions?
Furthermore, our purpose is to find two nat numbers such that so and so is satisfied. The epsilon language is hidden in this, because you say something lies between -k/n and k/n, but it is restricted to natural numbers.
Biggest problem with said definition is the non-strict inequalities.
Definition does not prevent existence of index N such that p(N)-A = k/N >0 [assuming natural numbers to be 1, 2, 3, ... as usual and if not we would already have a problematic definition (division by zero)].
Then what is to prevent p(n)-A=k/n>0 for all natural numbers. This poses a problem, because non-strict inequality is allowed.
No matter the choice of k or n, because k/n is positive. Does this, now, imply that a limit, if it exists, need not be unique? In a metric space, it's again proven, that a limit IS unique. (In a topological space a limit need not be unique, but that's another matter)

Elaborating further on limits. It is, in fact, NOT sufficient to check the validity of limit using only natural numbers, it only works the other way around. If we know a limit exists, then it May suffice to check some condition not for every epsilon >0, but only every n in N.