Young's inequality for convolutions in $l^2$

Suppose $a_n$ is a sequence of complex numbers such that $|a_n|$ is in $l^2$. Let$$c_m = \sum_{k=0}^m a_k a_{m-k}.$$I wonder if the following inequality holds$$\sum_{m=0}^\infty |c_m|^2 \leq \left(\sum_{m=0}^\infty |a_m|^2\right)^2.$$

This seems to be the discrete version of Young's inequality for convolutions, but I am not finding a proof or a good reference.

The inequality is false (see Lxm's comment below). Is the following true$$\sum_{m=0}^{\infty} \frac{|c_m|^2}{m+1} \leq \left( \sum_{m=0}^{\infty}|a_m|^2 \right)^2.$$