Say I have a sequence where for each positive integer n, $\displaystyle a_{n+1}=a_n- \frac {1}{n}$. Is there an $\displaystyle a_1$ I can choose which will make this a convergent sequence?

It seems to me that if one choice of $\displaystyle a_1$ will work, any choice will work, but that's just my intuition. I'm not sure how to go about proving it.