Prove that if the SUM of a_n converges then lim a_n = 0.?

when you state that sum converges to L you mean that lim {n---> ∞} S_n = L, where S_n = Σ {k=0,1,2,...n} a_k. But if S_n---> L then so does S_(n+1)---> L....but S_(n+1) = S_n + a_(n+1)---> L = L + lim{n--->∞} a_(n+1), or lim a_k---> 0 as k---> ∞