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I think this is a relatively straight-forward question. It is part of a larger proof that I am trying to do.

If I have an arbitrary, bounded sequence of functions in $L^2(\mathbb{R}^n)$, is any convergent subsequence of this bounded sequence uniformly convergent?

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No. Take a sequence of continuous functions that converge pointwise to a discontinuous function in a suitable way. E.g. (specialising to $L^2([0,1])$) we can use the sequence of $f_n$s whose graphs consist of three line segments: from $(0,0)$ to $(\frac12,0)$, from $(\frac12,0)$ to $(\frac12+\frac1{2n},1)$, and from $(\frac12+\frac1{2n},1)$ to $(1,1)$.

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