Let $S$ denote the vector space of all finitely nonzero sequences; that is, $X =(X_n) \in S$ if $X_n = 0$ for all but finitely many n. Show that $S$ is not complete under the sup norm $\| X \|_\infty = \sup_n |X_n|$.
What I have so far is that is $S$ is vector space of finitely non-zero sequence then $t_n(k)=1/2^k$ where $k=1, 2, 3$... I know that the norm is also the same as the norm of $L_\infty$. If I take the limit of $t_n$ as $k$ goes to infinity, then it converges to $0$. I think $\|t_n\|=1/i^2$. I am not sure how all of this tells me that $S$ is not complete under the particular Sup norm.
Try these $t_n(k) = 1/2^k$ if $0\le k \le n$ and $0$ otherwise. This is uniformly Cauchy but it does not converge in your vector space.