Showing that a function from Sobolev space $W^{1,2}$ decays to zero

Question

Given an unbounded real interval $I$, I would like to show that a function $f \in W^{1,2}(I)$ decays to zero as $|x|$ approaches $\infty$. Here are my attempts.

1. Without loss of generality, assume $I=(0,\infty)$. Choose $f\in W^{1,2}((0,\infty))\cap C^\infty((0,\infty))$ and $\eta_n\in C^\infty([0,2n])$ such that (i) $0\leq \eta_n(x)\leq 1$ for all $x \in (0,\infty)$, (ii) $|\eta_n'(x)|\leq \frac{2}{n}$ for all $x \in (0,\infty)$, (iii) $\eta_n(x)=1$ for all $x\leq n$, so $\lim_{n \to \infty}f\eta_n = f$ in $L^2((0,\infty))$.
2. $\|f'-(f\eta_n)'\|_{2}\leq \|f'-f'\eta_n\|_{2}+\frac{2}{n}\|f\|_{2}$, so $\lim_{n \to \infty}f\eta_n = f$ in $W^{1,2}((0,\infty))$.
3. If $f$ is compactly supported in $[0,n]$, then $|f(x)|\leq \|f'\|_{W^{1,2}}$.
4. So the inclusion $C_c^\infty([0,\infty))\to C_0([0,\infty))$ can be extended to continuous $W^{1,2}((0,\infty))\to C_0([0,\infty))$.

This construction was based on the similar case for $W^{1,1}((0,\infty))$. If I made any mistake or there exists any simpler way, kindly let me know.