Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

Question

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$.

I know that having the Bolzano-Weierstrass property means that every sequence $(x_n)_{n=1}^{\infty}\subset X$ has a convergent subsequence $x_{n_k} \xrightarrow{k\to\infty} x\in X$, but I'm unsure how to use this to prove $X$ is complete.

Answer 1

Let $\varepsilon>0$; then you know that you can find $r$ such that:

1. for every $m,n>r$, $\|x_m-x_n\|<\varepsilon/2$
2. for every $k>r$, $\|x_{n_k}-x\|<\varepsilon/2$

So, suppose $n>r$; then, for some $k>r$, you have $n_k>n$ and $$ \|x_n-x\|\le\|x_n-x_{n_k}\|+\|x_{n_k}-x\|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2} $$