Conditions of rudin's 7.24 theorem /existence of a uniformly convergent sequence but not equicontinuous

Question

this is theorem 7.24 of rudin's principle of mathematical analysis:

If $K$ is a compact metric space, if $f_n \in C(K)$ for $n=1,2,3,...$ ($C(K)$ being a set of complex-valued, continuous and bounded functions), and if $\{f_n\}$ converges uniformly on $K$, then $\{f_n\}$ is equicontinuous on $K$.

I was wandering about necessity of $K$ being compact.

can anyone help me with an example of a sequence of functions $\{f_n\}$ on $C(K)$( which $K$ is not a compact metric space) such that $\{f_n\}$ converges uniformly on $K$,but then $\{f_n\}$ is NOT equicontinuous on $K$.

Answer 1

Well, if $K$ isn't compact then its perfectly possible that $f_1$ is not uniformly continuous. If you add the condition that every function be uniformly continuous the theorem still holds.