I recently was trying to prove this statement: for any uniform Lipschitz constant $k$ and bound $M$
$A = \{ f : [0, 1] \to \mathbb{R} \ | \ f'(x) \leq k, |f(x)| \leq M \}$
is precompact in the supremum norm. I didn't know that there was a similar theorem called Arzela-Ascoli. My attempt was as follows:
- Let $S_0 = \{ f_i \in A\}$ be a sequence. For any $n > 0$, by Heine-Borel, there is a subsequence $S_n = \{ f_j \}$ of $S_{n-1}$ for which $(f_j(0), f_j(1/2^n), f_j(2/2^n), \dots, f_j(1)) \in \mathbb{R}^{2^n + 1}$ converges in the supremum norm as $j \to \infty$.
- Letting $S(i) = S_i(i)$ should be a convergent subsequence of $S_0$ since for every $\epsilon$ there is $N$ such that no $f_i(x)$ can differ from the nearest $f_i(k/2^N)$ by $\epsilon/2$ (equicontinuous - which follows from the uniform Lipschitz - is sufficient for this), and $N_2$ such that no $S(i)(k/2^N)$ can differ from $S(j)(k/2^N)$ by $\epsilon/2$ where $i, j > N_2$ due to how we constructed $S$.
After learning about Arzela-Ascoli, I'm not confident in my "proof" since the real proof is much more advanced (granted, I didn't attempt the converse). What are the gaps here?