Showing that the sequence is Cauchy if $|x_{n+1}-x_n|

Question

Let $a_n \ge 0$ for all $n\in \Bbb N$. Let $\{x_n\}$ be a sequence of real numbers such that $|x_{n+1}-x_n|<a_n$ for all $n\in \Bbb N$. If $\sum a_n$ is convergent then prove that $\{x_n\}$ is a Cauchy sequence in $\Bbb R$.

We have for any $p=1,2,\cdots$ that $|x_{n+p}-x_n|\le a_n+a_{n+1}+\cdots +a_{n+p-1}\le \sum_{k=1}^{n+p-1} a_k$. From here how can I show that $\{x_n\}$ is Cauchy ?

Any hint please?

Your final upper bound $\sum_{k=1}^{n+p-1} a_k$ has a needlessly bad lower summation index — Calvin Khor, Nov 09 '20 at 08:17
Same proof as in https://math.stackexchange.com/questions/182830/proving-that-a-sequence-such-that-a-n1-a-n-le-2-n-is-cauchy?rq=1 — Kavi Rama Murthy, Nov 09 '20 at 08:18
This question in some form appears too often. Did you try to find a duplicate? — Kavi Rama Murthy, Nov 09 '20 at 08:20

score 2 · Accepted Answer · answered Nov 09 '20 at 08:17

2

Hint: bound $a_n+\cdots+a_{n+p-1}\leq\sum_{k=n}^{\infty}a_k=\sum_{k=1}^{\infty}a_k-\sum_{k=1}^na_n$.

answered Nov 09 '20 at 08:17

Especially Lime

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Showing that the sequence is Cauchy if $|x_{n+1}-x_n|

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