What is the significance of the Uniform Cauchy Criterion vs just being uniformly convergent?

Question

So, there is a definition in my book for the uniform convergence of a sequence of functions:

Let $E$ be a nonempty subset of $\mathbb{R}$. A sequence of functions $f_n:E\to\mathbb{R}$ is said to converge uniformly on $E$ to a function $f$ if and only if for every $\epsilon$ there is an $N \in \mathbb{N}$ such that $n \ge N$ implies $|f_n(x) - f(x)| < \epsilon$ for all $x \in E$.

Now, the definition for the Uniform Cauchy Criterion is essentially the same, except that it states:

... Then $f_n$ converges uniformly on $E$ if and only if for every $\epsilon > 0$ there is an $N \in \mathbb{N}$ such that $n, m \ge N$ implies $|f_n(x)-f_m(x)| < \epsilon$ for all $x \in E$.

What is the signifance of this? What is so important about there being an $m \ge N$ such that $|f_n(x) - f_m(x)| < \epsilon$?

My book doesn't touch on this for some reason.

Answer 1

Let $X$ be a set endowed with a metric $d$. A sequence $(x_n)_{n\in\mathbb{N}}$ is convergent to $x\in X$ with respect to the metric $d$ if for every $\epsilon>0$ there exists an $N\in \mathbb{N}$ such that for all $n>N$, $d(x_n,x)<\epsilon$. Similarly, the sequence is called Cauchy if for every $\epsilon>0$ there exists an $N\in\mathbb{N}$ such that $n,m>N\implies d(x_n,x_m)<\epsilon$. In your case, $X$ is some set of functions from $E\to \mathbb{R}$, and the metric used is the sup-metric: $d(f,g)=\sup_{x\in E} |f(x)-g(x)|$. For any metric space, every converging sequence is Cauchy. However the converse does not always hold. Metric spaces for which this does, (i.e. spaces in which every Cauchy sequence is convergent) are called complete. If we assume in your case, that we work with all bounded functions from $E$ to $\mathbb{R}$, then this space is in fact complete, hence the two notions are equivalent. Beware, however, that this does not always hold for some more "unnatural" function spaces.

Answer 2

There are two distinct concepts involved here.

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The first is convergence of a sequence of real numbers (or any sequence of elements of a metric space, really, but let's ignore that for now) vs. the cauchy criterion. You call a sequence $(x_n)_{n\in\mathbb{N}}$ of $x_i \in X$ convergent if there's an $x \in X$ such that $$ \forall \varepsilon\; \exists N \; \forall n \geq N \; \|x_n-x\| \leq \varepsilon. $$ Note that this definition only makes sense if you have a suitable limit $x$ of the sequence $(x_n)_{n\in\mathbb{N}}$ at hand! Which sometimes can turn out to be a problem. For example, you might want to construct the reals as the limits of suitable sequences of rational numbers. But how would you formalize "suitable"? Intuitively, you'd want to look at all sequences which, somehow converge - but that's circular! Since we haven't yet defined the reals, the only set the potential limits of sequences can come from would be the rationals - but then, we'd only look at sequences of rationals which converge to a rational limit, which of course won't suffice to construct the reals.

The cauchy criterion solves this, by providing a way to define what "convergent" (note the quotes!) sequences are, without requiring the limit to (yet) exist. Instead of demanding that we can find, for every $\varepsilon$, and $N$ such that the values that come after $N$ lie closer than $\varepsilon$ to the limit, we instead require that any two values that come after $N$ lie closer than $\varepsilon$ to each other. Formally, this is stated as $$ \forall \varepsilon\; \exists N \; \forall n,m \geq N \; \|x_n-x_m\| \leq \varepsilon. $$

Note that both properties have the form $$ \forall \varepsilon\; \exists N \; \ldots $$

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Now let's look at uniformity vs. non-uniformity. Instead of sequences of values, we now look at sequences of functions $f_i \,:\, X \to Y$. Our goal is to somehow apply the concepts from above (or really any property of sequences of values of the form $\forall \varepsilon\;\exists N$) to sequences of functions. There are two possibilities.

The first is to apply the concepts pointwise. I.e., we pick an arbitrary $x \in X$, and look at the sequence of values $\left(f_n(x)\right)_{n\in\mathbb{N}}$. We can then asks whether all of these sequences converge, or satisfy the cauchy criterion, or whatever. So what we've done is created a property of sequences of functions of the form $$ \forall x \in X\;\forall \varepsilon\;\exists N\; \textrm{"Some statement about the values $f_n(x)$"} $$ This approach yields the pointwise convergence of a sequence of functions, or the pointwise cauchy criterion.

We can also, however, demand something stronger, picking $\varepsilon$ and $N$ first, and only then quantifying over the $x \in X$. In other words, we apply our original property to functions by re-writing it as $$ \forall \varepsilon\;\exists N\;\forall x \in X\; \textrm{"Some statement about the values $f_n(x)$"} $$ The difference being that after chosing a $\varepsilon$, we must now be able to find a single $N$ which "works" for all $x\in X$, whereas previously we were allowed to use different $N$ for different $x \in X$. This approach is what yields uniform convergence, or the uniform cauchy criterion.