= Uniformly continuous integrable functions vanish at infinity
If a <uniformly continuous function> $u:\mathbb R^d\to\mathbb R$ belongs to <Lp space> for some $1\leq p<\infty$, then $u(x)\to0$ as $|x|\to\infty$. Otherwise choose points escaping to infinity with $|u(x_j)|\geq2\theta>0$. <Uniform continuity> gives one radius $\rho>0$ on which $|u|\geq\theta$ around every $x_j$. Extract disjoint such balls. Their contributions to $\int|u|^p$ are each at least $\theta^p|B_\rho|$, a contradiction. The finite-$p$ condition is essential: the constant function one lies in $W^{1,\infty}$ and does not vanish at infinity.
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