Let $V$ be a finite dimensional vector space with a norm $| \ |,$ and let $d(x,y)=|x-y|$ be the corresponding metric. Show that the metric space $(V,d)$ is complete.
Do I start off the proof by saying let $( x_{n_1}, x_{n_2}, \ldots, x_{n_m})$ be a Cauchy sequence then show $\forall \epsilon >0: \exists N: \forall m: m, n \ge N: d \left({x_n, x_m}\right) < \epsilon$?
Best Answer
Close. The definition of a complete metric space is one where all Cauchy sequences converge. So, take an arbitrary Cauchy sequence $\{x_n\}$ and show that for all $\epsilon>0$ $\exists N\in\mathbb{N}$ such that $\forall n,m > N$, $|x_n-x_m|<\epsilon$ implies that $\{x_n\}$ converges. What you have written there is the definition of a Cauchy sequence. You want to use this definition to show that there is an $x \in V$ such that $|x_n-x|<\epsilon$. Hint: every Cauchy sequence is bounded.