Property of subsequences

Question

I need help or hints for the following property: Let $(X,d)$ be a metric space and $\{x_n\}$ a Cauchy sequence. For every sequence $\{\varepsilon_k\}$ of positive numbers there is a subsequence $\{x_{n_{k}}\}$ of $\{x_n\}$ so that

$$ d(x_{n_{k}},x_{n_{k+1}})<\varepsilon_k $$

for every $k\in \mathbb{N}.$

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We know that the sequence converges so the distance goes to zero anyway. Do I have to use Epsilon delta notation or is there another aproach?

Answer 1

To each $k$ there exists some $n_k$ with $$ n,m \ge n_k ~ \Rightarrow ~ d(x_n,x_m) < \varepsilon_k. $$ W.l.o.g. you can choose $(n_k)$ strictly monotone increasing. Then $n_{k+1} > n_k$, hence $d(x_{n_k},x_{n_{k+1}}) < \varepsilon_k$.

If $X$ is not complete it may happen that $(x_n)$ is not convergent. This does not affect the conclusion above.