Must continuous $H^1(\mathbb{R}^2)$ function tend to zero at infinity?

It does not hold for general functions $u \in H^{1}(\mathbb{R}^{2})$, even if they are assumed to be smooth. The reason lies in the following lemma:

Let $n \ge 2$. Then for any $x_{k} \in \mathbb{R}^{2}, \; \epsilon>0$ and $\delta>0$ there exists a radial smooth function $u_{k}$ such that:

1) $u_{k}(x_{k})=1$

2) $u_{k}(x)=0$ for $|x-x_{k}|>\delta$

3) $\int_{\mathbb{R}^{n}} |\nabla{u_{k}}|^2 \le \epsilon$

Now choose a sequence of points $x_{k}$ going to infinity, choose corresponding $\delta_{n}=\frac{1}{n^{2}}$ and $\epsilon_{n}=\frac{1}{n^{2}}$.

Let $u=\sum_{n} u_{n}$.

Then $u \in H^{1}(\mathbb{R}^{2})$.

But for any $r>0$ we find some point $x_{r}$ with $|x_{r}|>r$ such that $u(x_{r})=1$.