Does every point wise convergent $f_n$ has a uniformly convergent subsequence?

Question

as the question above states - could you prove or provide a counter example to the statement that every pointwise convergent function has a UC subsequence?
Also a bit unrelated but could you also recommend some explanatory videos on the subject similar to the ones prof Leonard has on YouTube? These subjects just don’t sit well with me.

Answer 1

A sequence of functions $f_n(x)$ is pointwise convergent to $f(x)$ over some domain $D$ if $$ \forall \epsilon>0\ \ ,\ \ \forall x\in D\ \ ,\ \ \exists N(x,\epsilon)\ \ , \ \ n>N(x,\epsilon)\implies |f_n(x)-f(x)|<\epsilon $$ and is uniformly convergent if $$ \forall \epsilon>0\ \ ,\ \ \exists N(\epsilon)\ \ , \ \ \forall x\in D\ \ ,\ \ n>N(\epsilon)\implies |f_n(x)-f(x)|<\epsilon $$ The sequence $f_n(x)=x^n$ over $D=(0,1)$ is pointwise convergent to $f(x)=0$ since $$ \forall x\in (0,1)\ \ ,\ \ n>\log_x \epsilon \implies |x^n-0|<\epsilon $$ but, there is no subsequence of $f_n(x)$ that is uniformly convergent.