I need to prove that this sequence is convergent:

(an) = (1 + 1/(sqrt(2) + 1/sqrt(3) + .... + 1/sqrt(n) - 2sqrt(n)). I know that this sequence is decreasing and bounded below. Therefore, it converges. However, I'm not sure how to show it's lower bounded. It seems to be bounded below by -2. I can show its decreasing.

Thanks for any help.