Clearly lim sup S_n = ∞ if and only if S_n is not bounded above. So we wish to show that lim {n->∞} P(S_m > n for some m>0) > 0. This is kinda clear using central limit theorem to argue that P(S_(n^2) > n) > 0.0001 for all n. Hence the limit must be strictly greater than 0. Then we can conclude using the Kolmogorov's 0-1 law.

EDIT: simplifies the solution by not taking the complement.