pointers on how I tried
Part 1: If X,Y are real sequences converging to 0, it is easy to show aX + bY, where a,b are real will also converge to 0. So set of S of all such seqences is a vector space.
Part 2: Attempt by contradiction. Assume the space is finite dimensional, get a basis and then construct a sequence which though converges to 0, cannot be a expressed as a linear combination of the basis we selected earlier. Hence proving the space is infinite dimensional.
As a further hint any n-dimensions vector space is isomorphic to
Hope it helps