I have a function $u_0(x) \in H^1(\mathbb{R})$. It is nonnegative, compactly supported, and has a mass $M>0$.
I have to prove that its first moment is finite.
My attempt: $\int_{\mathbb{R}} xu_0(x) \leq (\int_{\mathbb{R}}x^2)^\frac{1}{2} (\int_{\mathbb{R}} u^2_0(x))^{\frac{1}{2}}$
We know that the $u_0$ term is finite because it is in $H^1(\mathbb{R})$ but about the first term I am unsure. My guess was that instead of integrating over $\mathbb{R}$, we would consider the support of $u_0$ which is compact. Thus, the first term evaluated over a compact support is finite, and that's enough?