$\frac{a_n}{1+a_n} \to l \neq 1$ and $a_n$ bounded implies that $a_n$ converges

The question is essentially contained in the title. Suppose $a_n$ is a bounded sequence and $\frac{a_n}{1+a_n} \to l \neq 1.$ I want to show that $a_n$ converges.

Of course, assuming it does converge, it is easy to see that $a_n \to \frac{l}{l-1}.$ However, I am not sure how one can use the boundedness condition to see that $a_n$ converges (I am also not sure how to find a counterexample in case $a_n$ is not bounded!).

Thank you.

Hint: if $b_n = a_n/(1 + a_n)$, then $a_n = b_n/(1-b_n)$.
By the way, the condition that $a_n$ is bounded is not needed here (it would be needed if you weren't told $\ell \ne 1$).
• So then, $a_n$ is a quotient of convergent sequences, and thus converges. But how are we making use of the boundedness assumption? – user127519 Feb 10 '14 at 0:38
• If we weren't told $\ell \ne 1$, $a_n$ bounded could be used to prove that $\ell \ne 1$. – Robert Israel Feb 10 '14 at 2:30
Hint: Define $$f(x) = \frac{x}{1-x}$$ Noting that $f(x)$ is continuous except at $x = 1$. Suppose that $b_n = \frac{a_n}{1+a_n} \to l \neq 1$, what can we say about the sequence $f(b_n)$?