Being given a real sequence $(U_n)_n$ $$Un \to l \in R$$ And considering a sequence $(f_n)_n$ of continuous functions such as $$f_n(U_n) \to l' \in R$$ Does if follow that $$ f_n(l) \to l' $$
I tried proving this using another variable $m$ but I don't really know what are the legal operations I can do with that. Here is some suggestion that seems reasonable in the case of continuous functions (but here again, can't prove it):
Is it legal to do : $$\lim_{n \to \infty} f_n(U_n) = \lim_{n \to \infty} \lim_{m \to \infty} f_n(U_m)$$
Thanks <3
It does not follow (it is simply not true). Pick the sequence $U_n=1/n$, then $l=0$. Now pick a sequence of continuous functions $(f_n)$ such that $f_n(0)=1$ and $f_n(x)=0$ for $x>1/(2n)$ (construct such functions!). Then $f_n(U_n)=0$ and $f_n(l)=f_n(0)=1$ and thus you will not get equality of your limits.