For a random variable $X$ on $[0, 1]$ with $F(1) = 1$ and $F(x) < 1$ for all $x < 1$, show that $E(|X|^r)^{\frac{1}{r}} \to 1$ as $r → ∞$. If $F$ is such that $F(x) < 1$ for all $x ∈ R$ and all moments exist, then show $E(|X|^r)^{\frac{1}{r}} \to \infty$ as $r → ∞$.
Can someone provide a hint?