= Solution
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 $|x_j|\leq M$ for all $j$ and $x\in\ell^\infty$. Given $\varepsilon>0$, choose $N$ such that $\sup_j|x_j^{(n)}-x_j^{(m)}|<\varepsilon$ for $m,n\geq N$. Letting $m\to\infty$ in each coordinate gives $\sup_j|x_j^{(n)}-x_j|\leq\varepsilon$ for $n\geq N$. Thus $x^{(n)}\to x$ in norm, and \b[$\ell^\infty$ is a <Banach space>.]
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