 Solution

No. Choose a nonzero $\phi \in C_c^{\infty }(\mathbb{R})$ and a number $L$ larger than the diameter of its support, and setTranslation preserves both the $L^2$ norm and the derivative norm, so $(u_n)$ is bounded in $H^1(\mathbb{R})$. Distinct $u_n$ have disjoint supports, and henceThus the sequence has no $L^2$-Cauchy subsequence and therefore no strongly convergent subsequence. The embedding $H^1(\mathbb{R})\hookrightarrow L^2(\mathbb{R})$ is not compact.