= Solution
Choose a nonzero $\phi\in C_c^\infty(\mathbb R^n)$ supported in a ball of radius $1$, and let
$$
\phi_k(x)=\phi(x-3ke_1).
$$
Translation preserves both the $L^2$ norm and the weak-derivative norms, so
$$
\lVert\phi_k\rVert_{H^1}=\lVert\phi\rVert_{H^1}
$$
for every $k$. Thus $(\phi_k)$ is bounded in the <Sobolev space> $H^1(\mathbb R^n)$.
Distinct translates have disjoint supports. Hence, for $j\ne k$,
$$
\lVert\phi_j-\phi_k\rVert_2^2
=\lVert\phi_j\rVert_2^2+\lVert\phi_k\rVert_2^2
=2\lVert\phi\rVert_2^2.
$$
No subsequence is Cauchy in $L^2$, so no subsequence converges there. The embedding $H^1(\mathbb R^n)\hookrightarrow L^2(\mathbb R^n)$ is therefore not a <compact operator>. This is the <noncompactness of a Sobolev embedding by translation> caused by <loss of compactness at infinity>.
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