$1($a$)(3$ points$)$

Let $\{a_n{\}}_n=1^{}$ be a sequence with positive terms and suppose the $N$ th partial sums of the series ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ satisfy $S_{N}1$AAN$1.$ Prove ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ converges.