I have an exercise which asks to prove or provide a counterexample to the following claim:
Given $f\in W^{1,2}(\mathbb{R}^2)$, there exists a radius $R>0$ such that $\vert f(x)\vert \leq 1$ for every $x\in \mathbb{R}^2\setminus B_R(0)$, where $B_R(0)$ denotes the ball of radius $R$ and centered at the origin.
We have seen that $W^{1,2}(\mathbb{R}^2)=W^{1,2}_0(\mathbb{R}^2)$, where the latter is the closure (w.r.t the $W^{1,2}(\mathbb{R}^2)$ norm) of smooth and compactly supported functions on $\mathbb{R}^2$, so intuition would suggest that functions in $W^{1,2}(\mathbb{R}^2)$ "decay at infinity", but I don't know how to formally prove that $L^p$-convergence of a sequence of compactly supported smooth function implies that the limit function is zero at infinity.
Or if the claim is false, I can't find a counterexample.
Thanks in advance for your help.