Critique my proof of the Arzela-Ascoli Theorem (with one question)

[jdinatale]

In part (b), I'm not sure why it's important that r_1, r_2, ..., r_n is finite. Any thoughts? One thing I'm concerned about is if I have really shown that g_k converges uniformly because I did not shown that the N chosen was independent of the x.

 

[micromass]

The problem is of course that every $r_i$ gives rise to a different N.

That is: you know that $(g_s(r_i))_s$ is Cauchy, thus we can write:

$$\forall \varepsilon >0:\exists N_i: \forall s,t>N_i:~|g_s(r_i)-g_t(r_i)|<\varepsilon$$

I wrote $N_i$ here instead of $N$ because we do not have only one N.

Now you must combine the $N_i$ into one N. This will use finiteness.


A critique of your proof: you have not shown that we can actually choose a finite set $\{r_1,...,r_n\}$ that satisfies the criteria. This is very important and uses something essential.

Furthermore, your proof of (c) isn't quite nice. You say "we may choose $\delta>0$", but you do realize that this delta was already chosen in (b)?