State and prove Hölder's Inequality.

[Jensen's inequality, and other standard results, may be assumed.]

Let $\left(X_{n}\right)$ be a sequence of random variables bounded in $L_{p}$ for some $p>1$. Prove that $\left(X_{n}\right)$ is uniformly integrable.

Suppose that $X \in L_{p}(\Omega, \mathcal{F}, \mathbb{P})$ for some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and some $p \in(1, \infty)$. Show that $X \in L_{r}(\Omega, \mathcal{F}, \mathbb{P})$ for all $1 \leqslant r<p$ and that $\|X\|_{r}$ is an increasing function of $r$ on $[1, p]$.

Show further that $\lim _{r \rightarrow 1^{+}}\|X\|_{r}=\|X\|_{1}$.