Proof of Series Vanishing Condition

Proof

1. Suppose $\sum a_n$ converges to $s\in \mathbb{R}$
2. Then, by the definition of convergence, ${\lim }_{n\to \infty }{S}_n=s$
3. WTS: ${\lim }_{n\to \infty }{a}_n=0$
4. Then, consider ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }(a_n+0)$
5. $={\lim }_{n\to \infty }(a_n+a_1+\cdots +a_{n-1}-a_1-\cdots -a_{n-1})$
6. $={\lim }_{n\to \infty }(S_n-S_{n-1})$ by Series Partial Sum
7. $={\lim }_{n\to \infty }(S_n)-{\lim }_{n\to \infty }(S_{n-1})$ by Limit Properties
8. $={\lim }_{n\to \infty }(S_n)-{\lim }_{n\to \infty }(S_n)$ as ${\lim }_{n\to \infty }(n-1)\sim n$
9. $=s-s$
10. $=0$
11. $\square$