I've been going at a statement trying to prove it or find a counterexample:
$f_n : R\rightarrow R$ is a sequence of bounded functions converging uniformly to $f: R \rightarrow R$. If each function $f_n$ has a limit at infinity, then $f$ has a limit at infinity.
I'm pretty sure the statement is true because since $f_n$ is converging uniformly to $f$ then every behaviour from $f_n$ should be in $f$ but not sure how to prove that. Any hints or am I totally wrong to assume that?