I want to prove that:
If the continuous function $f(x)$ has a bounded limit as $x$ goes to $\infty$ i.e $$0<L=\liminf (f)\leq S=\limsup(f)<\infty,$$ then for every $x_0 \in [L,S]$ there exists a sequence $x_n$ and $x_n$ goes to $\infty$ as $n$ goes to $\infty$ such that $$\lim\limits_{n \rightarrow \infty} f(x_n)=x_0.$$
If $x_0\in\{L,R\}$, the result is trivial. Otherwise, there is $\varepsilon>0$ such that $x_0\in[L+\varepsilon,R-\varepsilon]$. For any $n>0$ there exist $a,b>n$ such that $f(a)<L+\varepsilon$, $f(b)<R-\varepsilon$. Use Intermediate value theorem to show that, for some $c>n$, one has $f(c)=x_0$.