Let $f$ be a continuous and decreasing function and $\{a_n\}$ sequence such that $a_n \rightarrow \infty$ and $lim_{a_{n}\rightarrow\infty} f(a_n) = a$. Then $lim_{x\rightarrow\infty} f(x) = a$.
I am at a loss. I tried to prove it by contradictory, but I failed so far.
Hints: There is a subsequence $a_{n_k}$ which strictly increases to $\infty$. For $a_{n_k} \leq x \leq a_{n_{k+1}}$ we have $f(a_{n_k}) \geq f(x) \geq f(a_{n_{k+1}})$. Note that $f(a_{n_k}) \to a$ and $f(a_{n_{k+1}}) \to a$. Can you finish?