How to prove that $\sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}}$ converges absolutely If $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, show that
$$\sum_{n=1}^{\infty}\dfrac{a_{n}}{1+a_{n}}$$ converges absolutely.

My try: since $\sum_{n=1}^{\infty}a_{n}$ converges absolutely, then
$$\sum_{n=1}^{\infty}|a_{n}|$$ converges, then there exsit $M&gt;0$, such
$$\sum_{n=1}^{N}|a_{n}|&lt;M$$

then, how to prove than
$$\sum_{n=1}^{N}\dfrac{|a_{n}|}{|1+a_{n}|}&lt;cM?$$
where $c$ is a constant?
Hint: Prove that if $\sum a_n$ converges absolutely and $b_n$ is a bounded sequence, then also $\sum a_nb_n$ converges absolutely.
It is well-known that 
$$
\frac{x}{x+1}&lt;x
$$
for positive x. Therefore we have for each term that $|a_n|\geq \frac{|a_n|}{|1+a_n|}$. Therefore 
$$
\infty &gt;\sum_{n=0}^\infty |a_n| \geq \sum_{n=0}^\infty \frac{|a_n|}{|1+a_n|}
$$
And the sum converges absolutely.