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?