Pointwise & uniform boundednes

[camillio]

In Baby Rudin, Theorem 7.25 states:
If K is compact, f_n \in C(K) for n=1,2,3,... and if {f_n} is pointwise bounded and equicontinuous on K, then
(a) {f_n} is uniformly bounded on K

The theorem continues with point (b), which I understand.

My question is, whether point (a) needs the equicontinuity, since:
* C(K) is a set of continuous bounded functions on a compact space, i.e. uniformly continuous, hence there is no function g s.t. g(x) \to \infty.
* if the previous holds, then by the hypothesis of pointwise boundedness of \{f_n\}, there must exist a number \sup_{x \in K, n=1,2,3,...} |f_n(x)| = M<\infty, x\in K arbitrary and hence the uniform boundedness should hold.

Am I wrong?

 

[micromass]

The sequence

f_n:[0,1]\rightarrow \mathbb{R}:x\rightarrow \frac{nx}{x^2+(1-nx)^2}

seems to be a counterexample to your claim. See example 7.21 of Rudin, I modified it a bit.

 

[camillio]

Thanks a lot! I assume, that the reason is that f_n converges pointwise for every x\in K, but due to the points 1/n in your example, it is impossible to find a global finite M s.t. |f_n(x)| \leq M for all x \in K. Right?

I really do not understand why usually things go quite well, but at some moments I get stuck with (obvious?) things :(