Prove $M_{m\times n}(\mathbb{R})$ is complete

Question

I'm struggling with an exercise in my real analysis course. The question it asks is:

Prove that $M_{m\times n}(\mathbb{R})$ with the metric induced by $\left\Vert \cdot \right\Vert_{\infty} $ is complete. (Where $\left\Vert M \right\Vert_{\infty} :=\max\{m_{ij}\,|\,1\leq i \leq n, 1\leq j \leq n\} $

Edit note: I am generally struggling with this concept as a whole, so also any advice on how to tackle problems in this topic of a similar nature would be very much appreciated

Answer 1

Let $\bigl(M(n)\bigr)_{n\in\mathbb N}$ be a Cauchy sequence of matrices. Then, for each $i$ and each $j$ in $\{1,2,\ldots,n\}$, each sequence $\bigl(M(n)_{ij}\bigr)_{n\in\mathbb N}$ is a Cauchy sequence of real numbers, since$$\bigl|M(m)_{ij}-M(n)_{ij}\bigr|\leqslant\bigl\|M(m)-M(n)\bigr\|$$for each $m,n\in\mathbb N$ and since $\bigl(M(n)\bigr)_{n\in\mathbb N}$ is a Cauchy sequence. Let $M_{ij}=\lim_{n\to\infty}M_{ij}(n)$ and let $M=(M_{ij})_{1\leqslant i,j\leqslant n}$. Then$$\lim_{n\to\infty}M_n=M.$$