\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(f\) be continuous on \([a, b]\) with \(\|f\|_\infty = M\). Show that for every \(n \in \mathbb{N}^*\), there exists a step function \(\varphi\) on \([a, b]\) with \(\|\varphi\|_\infty \le M\) and \(\|\varphi - f\|_\infty < 1/n\).
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