Let $X_n$ approach $X$ in probability as $n$ goes to $\infty$. I want to show that if $\sup E |X_n|^p$ is finite for $p > 1$, then there is also convergence in $\mathcal{L}^q$, $q < p$.
I have managed to show that $E |X|^p$ is also finite, but am now stuck.
Fix $1 \leq q < p$. Note that
$$\|X_n-X\|_{L^q}^q = \int_{|X_n-X| \leq \epsilon} |X_n-X|^ q\, d\mathbb{P} + \int_{|X_n-X| > \epsilon} |X_n-X|^q \, d\mathbb{P} \tag{1}$$
for any $\epsilon>0$ and $n \in \mathbb{N}$. Obviously,
$$\int_{|X_n-X| \leq \epsilon} |X_n-X|^q \, d\mathbb{P} \leq \epsilon^q.$$
In order to estimate the second term on the right-hand side of $(1)$, note that, by Hölder's inequality,
$$\int_{|X_n-X| > \epsilon} |X_n-X|^q \, d\mathbb{P} \leq \mathbb{P}(|X_n-X| >\epsilon)^{p/(p-q)} \cdot \left(\int |X_n-X|^{p} \, d\mathbb{P} \right)^{q/p}.$$
Now use the inequality
$$|X_n-X|^{p} \leq 2^{p} (|X_n|^{p}+|X|^{p}),$$
the fact that $X_n$ converges to $X$ in probability and $\sup_n \|X_n\|_p+\|X\|_p<\infty$.
Remark: This is a particular case of Vitali's convergence theorem.
