= Solution
On a bounded Lipschitz domain $\Omega\subset\mathbb R^d$, the <Rellich-Kondrachov compactness theorem> states that $W^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$ is compact for $1\leq p<\infty$ and
$$
\frac1q>\frac1p-\frac1d,
$$
with $1/q=0$ for $q=\infty$. Thus $q<p^*$ when $p<d$, every finite $q$ is allowed when $p=d$, and the embedding into continuous functions with their uniform norm is compact when $p>d$.
For the main proof, a <Sobolev extension operator> puts a bounded sequence into a common compactly supported region of $\mathbb R^d$. The <Sobolev fundamental theorem of calculus on lines> gives the uniform translation estimate $\|u(\cdot+h)-u\|_p\leq|h|\|\nabla u\|_p$. Convolution with a <mollifier> therefore approximates the sequence uniformly in <Lp space>. At any fixed smoothing scale, its derivatives and supremum are uniformly bounded, so the <Arzelà-Ascoli theorem> gives a convergent subsequence. A diagonal choice and the uniform approximation give strong <Lp space> convergence. The finite measure of $\Omega$ handles $q<p$, while <Lp interpolation inequality> with the bounded Sobolev embeddings upgrades this to the larger subcritical finite <Lp spaces>; for $p>d$, <Morrey's inequality> gives uniform equicontinuity directly and the <Arzelà-Ascoli theorem> gives uniform convergence. Boundedness of the domain is essential.
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