Please write out your questions properly. What you've written doesn't make any sense. Is 0n supposed to be 0*n, for n in N? It clashes with what you mean by using Sn, you see. It is not right to say each n is finite though arbitrarily large. Each n is not arbitrarily large, each n is just an integer, and doesn't vary at all. There is no need to invoke 'finiteness of n'. 0*n=0 for all n in N. That is just a simple fact, so S_n=0 for all n. Constant sequences obvisouly converge.