I'm trying to understand how to proof Cauchy sequence that converges.
Given that let $a_i$ be a sequence of number such that $a_i \in \{-1, 0,1,\}$ for each i.
Let $s_n$ be a sequence that $s_n = \frac{a_1}{3} + \frac{a_2}{3^n} + \frac{a_3}{3^3} + ... + \frac{a_n}{3^n}$.
Proof:
Suppose $s_n$ converge to $s$, where $\lim s_n = s$. Let $\epsilon > 0$ then there exist $N \in \mathbb{N}$ such that $|s_n - s| < \frac{\epsilon}{2}$. Then for all $n, m \geq N$, we have $$|s_n - s_m| = |s_n - s + s - s_m| \leq |s_n - s| + |s - s_m|$$ $$= |\frac{a_n}{3^n} - s| + |\frac{a_m}{3^m} - s| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$
Hence, we proved that $s_n$ is a Cauchy sequence. Therefore, the following sequence $s_n$ converges.
Can someone verify that if I did the proof correctly?