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Consider the subset S = $L^\infty(\mathbb{R}) \cap L^2(\mathbb{R})$ of $L^\infty(\mathbb{R})$. We know that $L^\infty(\mathbb{R})$ is a Banach space and complete. Is this subset $S$ complete under the $\|\cdot\|_\infty$ distance metric? If yes, where can I find a proof for it?

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