Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 22H / a

The supremum norm satisfies positivity, absolute homogeneity and the triangle inequality coordinatewise. Let $x^{(n)}$ be a Cauchy sequence in this norm. Each coordinate is Cauchy in $\mathbb{C}$ and has a limit $x_j$. The sequence of norms is bounded, so $|x_j|\le M$ for all $j$ and $x\in {\ell }^{\infty }$. Given $\epsilon >0$, choose $N$ such that ${\sup }_j|x_j^{(n)}-x_j^{(m)}|<\epsilon$ for $m,n\ge N$. Letting $m\to \infty$ in each coordinate gives ${\sup }_j|x_j^{(n)}-x_j|\le \epsilon$ for $n\ge N$. Thus $x^{(n)}\to x$ in norm, and ${\ell }^{\infty }$ is a Banach space.