Hey guys. I think this is a nonproof since the proof given in baby Rudin is a bit longer, but I'm not sure where it breaks down. The well-known theorem is:
Suppose converges uniformly to on a set E in . Let x be a limit point of E and suppose that .
Then converges, and .
Denote by . For arbitrary n, note that by continuity of . Given , choose so that for , for all . Since this holds for all , it holds for the limit as well, i.e. so that if . QED.
If that's wrong, I need to know since I have the feeling it would indicate some big flaw in my understanding. If it's okay, I think the issue might be that the proof doesn't generalize to arbitrary metric spaces perhaps?
Oh, wow, you're right. For some reason I thought it was a given.
The proof given makes a lot more sense now. The strategy seems to be to show that converges first, then use that to show that the limit exists, and that they are equal.
Heh, that's actually kind of embarrassing.