I am learning about sobolev spaces and try to prove some simple properties. Our definition of $H^s(\mathbb{R}^n)=W^{k,2}(\mathbb{R}^n)$ is the following
$$ f \in H^s \text{ if and only if there exists a sequence } f_k \in C_0^{\infty}(\mathbb{R}^n), \text{ such that } \Vert f_k-f\Vert_{H^s} \rightarrow 0 $$ We defined the norm for the space $H^s$ to be $$ \Vert f \Vert_{H^s} = \Big(\int_{\mathbb{R^n}}{(1+\vert y \vert^2)^s \vert \hat f(y) \vert^2dy}\Big)^{1\diagup2},$$ where $\hat f$ is the fourier transform of $f$.
For $s > \frac{n}{2}.\; $Let $f \in H^s(\mathbb{R}^n)$, than $f$ vanishes at $\infty$
I know that if $f_k \in C_0^{\infty}(\mathbb{R}^n)$ converges uniformly to $f$, than $f$ vanishes at $\infty$. So my "idea" would be to show that convergence in $\Vert \cdot \Vert_{H^s}$ for $s > \frac{n}{2}$ implies uniform convergence. Would appreciate any hint/help to prove the statement above using only the definition I have given.
This is very much related to a common Sobolev embedding theorem, namely that $H^s$ embeds continuously into $C^k\cap L^\infty$ if $s>k+n/2$ (for you, $k=0$).
Observe that if $u\in H^s$, then $$\int|\hat{u}(\xi)|\, d\xi=\int|\hat{u}(\xi)|(1+|\xi|^2)^{-s/2}(1+|\xi|^2)^{s/2}\, d\xi\leq \left(\int (1+|\xi|^2)^{-s}\, d\xi\right)^{1/2}\|u\|_{H^s}<\infty.$$ Hence, $\hat{u}\in L^1,$ then an application of the Riemann-Lebesgue lemma yields that $u$ vanishes at infinity.