Fix a function $\phi \in C_c(\mathbb{R}), \phi\not\equiv0,$ and consider the family of functions
$\mathcal{F} = \{\phi_n:n\in\mathbb N\},$ where $\phi_n(x) = \phi(x+n), x\in\mathbb{R}.$
The problem is to prove that $\mathcal{F}$ does not have compact closure in $L^p(\mathbb{R}) (1\le p <\infty),$ but there is a theorem that the closure $\mathcal{F}|_{\Omega}$ in $L^p(\Omega)$ is compact for any finite-measure subset $\Omega\in\mathbb{R}.$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $\mathbb{R}.$
I think I need to induce contradiction assuming $\mathcal{F}$ has compact closure in $L^p(\mathbb{R}),$ but I cannot come up with the next step.
Any hint or idea would be appreciated. Thanks in advance.
Hint: We have $\operatorname{supp}\phi\subset [-N,N]$ for some $N\in\mathbb N$. Can the sequence $\{\phi_{nN}\}_{n\in\mathbb N}$ have a convergent subsequence?