If $\{s_n\}_{n=1}^\infty$ is a sequence in $\mathbb R$ that satisfies $|s_{n+1} - s_n| \to 0$ then $s_n$ is cauchy.

Question

True/False: If $\{s_n\}_{n=1}^\infty$ is a sequence in $\mathbb R$ that satisfies $|s_{n+1} - s_n| \to 0$ then $s_n$ is cauchy.

I know for a sequence to be cauchy, $\forall \epsilon \gt 0$ $\exists$ positive integer $N$ such that for $n,m \in \mathbb N$ we have $|s_n-s_m| \lt \epsilon$ I'm not sure how to deal with the $|s_{n+1} - s_n| \to 0$.

Answer 1

Hint : Consider the sequence $s_n = \sum_{k=1}^n \frac1k$.

Answer 2

This is false, let $s_n=\log n$. Then $$s_{n+1}-s_n=\log\frac{n+1}n.$$

So, $|s_{n+1}-s_n|\to 0$, but $s_n\to\infty$.