Show that

$\displaystyle a_n=(\sum_{k=1}^\infty \frac{1}{k})-log(n+1)$ converges to some real number.

The problem hints that I'm supposed to use the fact that

$\displaystyle \sum_{k=1}^\infty \frac{x}{k(x+k)}$ converges uniformly on [0,1], which I've already proved.

I can't figure out how to connect these two facts at all... any help?