The unbounded domains do not admit compact imbeddings into $L^P$

Question

The unbounded domains do not admit compact imbeddings $W^{1,p}$ to $L^p$

For this ..i consider an example

$\Omega=\mathbb{R}$ , let $I=(0,1)$ and $I_j=(j,j+1)$ for $j\in \mathbb{Z}$

Let $f\in C^1$ function compactly supported in $I$

define $f_j(x)=f(x-y),j\in \mathbb{Z}$

clearly $f_j\in W^{1,p}(\mathbb{R})$ for any $1\le p < \infty$ and the sequence $\{f_j\}$ bounded in that space

now consider $|f_i-f_i|_{0,p,\mathbb{R}}=\int_{\mathbb{R}} |f_i-f_j|^p\; dx$ from we here how to processed

MY question

how to prove this sequence cannot have convergent subsequence