The $L^p(\mathbb R)$ norm is increasing as a function of $p$ (Update: It's false!) Update: This is false. See the answers for a counterexample.

Let $C\ge 1$ be a constant. Fix $f\in L^p(\mathbb R)$ for $p\ge C$. Show that $$p\rightarrow \left( \int |f|^p
 \right)^{1/p}$$ is non-decreasing.

Comments: I'm posting this because there is (surprisingly) no good reference for this fact on the internet. If I recall correctly, differentiating with respect to $p$ will do the trick. 
For the same problem on a finite measure space, see here.

This is not true: Consider the function $f(x)=\chi_{\mathbb{R}\setminus(-1,1)}(x)|x|^{-1/2}$. Then for $p\leq 2$ we have $\| f\|_{p}=\infty$ and for $p>2$ we get 
$$
\| f\|_p = \left(\frac{4}{p-2}\right)^{1/p}.
$$
