Let $s>n/2, \; f \in W^{s,2}(\Bbb R^n)$ . Then how can I show that there is an embedding into the space of uniformly bounded, continuous functions, that is, $$ |f(x)| \leqslant C\| f \|\_{W^{s,2}}$$ for almost all $x \in \Bbb R^n$ ? 


I think the general Sobolev embedding theorem cannot be applied in this case because the domain is $\Bbb R^n$ which is open. 



About Sobolev embedding theorem of the case $W^{s,2}$.
This is very standard, but perhaps buried in fancier things:


The Fourier transform $\hat{f}$ of $f$ is in $L^2$ for the weight $(1+|x|^2)^s$. Since $s>n/2$, the constant function $1$ is in that weighted $L^2$ space, so by Cauchy-Schwarz-Bunyakowsky,
$\int\_{\mathbb R^n} |\hat{f}| = \int |\hat{f}|\cdot 1\le |\hat{f}|\_s \cdot |1|\_t$, 
where $t=-s$ [to avoid TeX bug?] and the subscript denotes the weight. Thus, $\hat{f}$ is in $L^1$. Thus, by Fourier inversion and Riemann-Lebesgue, $f=\hat{\hat{f}}$ is uniformly continuous and goes to $0$ at infinity.


