Let $I_\alpha=\int_{(0,\infty)} \frac{1}{x^\alpha} \ d\mu(x)$, where $\mu$ is the Lebesgue measure. Prove that $I_\alpha\in (0,\infty)$ for $\alpha>1$.
I considered using DCT to tackle this problem, but I got stuck on choosing function $g(x)$ such that $|\frac{1}{x^\alpha}| \leq g(x)$ and $g(x)$ is integrable. In any case, I suspect that DCT is useless for this problem. Can you provide hints on how to prove this?