# (False?) Corollary to Arzela-Ascoli

Note: A corollary of Arzela-Ascoli Theorem asks a similar question, but this question assumes pointwise convergence. Here we do not assume this.

(A version of) the Arzela-Ascoli theorem reads

If $$\{f_n\}_n$$ is a sequence of functions on a metric space $$C$$ which is uniformly bounded and uniformly equicontinuous, then there exists a uniformly convergent subsequence $$\{f_{n_k}\}_k$$.

Does this not mean that the entire sequence of functions $$\{f_n\}_n$$ in the Arzela-Ascoli theorem, not just a subsequence? Here is why I think this might be plausible:

Let $$\{f_n\}$$ satisfy the hypotheses of Arzela-Ascoli. Uniform convergence of $$\{f_n\}_n$$ is equivalent to the convergence $$\begin{equation*} g_n := \sup_{x \in C} |f_n(x) - f(x)| \to 0 \qquad (n \to \infty) \, . \end{equation*}$$

Let $$\{g_{n_k}\}_k$$ be a subsequence achieving $$\limsup g_n$$, that is $$\begin{equation*} g_{n_k} \to \limsup g_n =: \ell \qquad (k \to \infty) \, . \end{equation*}$$

The corresponding subsequence $$\{f_{n_k}\}_k$$ remains uniformly bounded and uniformly equicontinuous, so Arzela-Ascoli provides a further subsequence, denoted $$\{f_{n_{k'}}\}_{k'}$$, which converges uniformly, i.e., $$\begin{equation*} g_{n_{k'}} \to 0 \, . \end{equation*}$$

Since $$\{g_{n_{k'}}\}_{k'}$$ is a subsequence of the convergent sequence $$\{g_{n_k}\}_k$$ (with limit $$\ell$$), $$\begin{equation*} \ell = \lim_{k' \to \infty} g_{n_{k'}} = 0 \, . \end{equation*}$$

Now $$\{g_n\}_n$$ is nonnegative and its $$\limsup$$ is zero, so $$g_n \to 0$$, which means the entire sequence $$\{f_n\}$$ uniformly converges to $$f$$.

I feel like I'm missing something since I've never seen Arzela-Ascoli stated as $$\{f_n\}$$ converges uniformly,'' merely that a subsequence converges. If that's the case, what am I missing? Do we need some sort of pointwise convergence of $$\{f_n\}_n$$ as well, as mentioned in the note?

• This is about (sequential) compactness or pre-compactness in a certain topology. Then the issue is easily illustrated in familiar (locally compact) metrics spaces such as closed balls in finite-dimensional Euclidean spaces. – paul garrett Feb 27 at 16:50
• I see. Do you mean to say that the issue is that in writing the limit $f$ above, that $f$ is merely a subsequential limit? Thus, $f_{n_{k'}}$ does not necessarily converge to $f$. It seems also that if we have some sort of convergence on $\{f_n\}$ (not just pointwise), such as weak, $L^p$, or in measure, that is implied by uniform convergence, then one can replace the pointwise-convergence assumption with that. If so, then I'm happy with that; I had originally asked this question to understand Lemma 1 of jstor.org/stable/2241171?seq=1#metadata_info_tab_contents. Thank you! – grogTheFrog Feb 27 at 18:52