Let $1\le p<\infty$ and $f\in C^{\infty}(\mathbb{R})$ be such that its all derivatives $f^{(n)} \in L^p(\mathbb{R})$ for every $n\in \mathbb{N} \cup \{0\}$. 
Is it true that $f\in L^\infty(\mathbb{R})$?
If it is true, do we have a generalization in $\mathbb{R}^n$? 

My partial answer for $p=1$:

For $p=1$ and $t\in \mathbb{R}$, we have
$$
|g(t)|=
|\int\_{0}^{t} g'(x) \ dx+g(0)|
\le
\int\_{0}^{t}
|g'(x)| \ dx
+|g(0)|
\le 
\|g'\|\_{L^1}
+|g(0)|.
$$ 
Thus, $g\in L^\infty(\mathbb{R})$ with 
$\|g\|\_{L^\infty} \le \|g'\|\_{L^1}+|g(0)|$.

Thanks for help and hint.