Let $(X_{n})_{n\in \mathbb N}$ be some random variables such that $\sup\limits_{n \in \mathbb N}E[\lvert X_{n}\rvert ^{q}]<\infty$ and $X_{n}\xrightarrow{\mathbb P}X.$
Show that $X_{n} \xrightarrow{L^{p}}X$ for $p < q$
My idea:
By Fatou's Lemma, since convergence in probability implies a.s. convergence of a subsequence then w.l.o.g let $(X_{n})_{n \in \mathbb N}$ be the sequence.
Then $$E[\lvert X\rvert^{q}]=E[\liminf\limits_{n \to \infty}\lvert X_{n}\rvert^{q}]\leq \liminf\limits_{n \to \infty}E[\lvert X_{n}\rvert^{q}] \leq \sup\limits_{n \in \mathbb N} E[\lvert X_{n}\rvert ^{q}]<\infty$$
and thus $\sup\limits_{n \in \mathbb N}E[\lvert X-X_{n}\rvert^{q}]<\infty$
Furthermore let $\varepsilon > 0$ and by Hölder:
$$E[\lvert X-X_{n}\rvert ^{p}]=E[\lvert X-X_{n}\rvert ^{p},\lvert X-X_{n}\rvert\leq \varepsilon]+E[\lvert X-X_{n}\rvert ^{p},\lvert X-X_{n}\rvert> \varepsilon]\\ \leq \varepsilon^{p}+E[\lvert X-X_{n}\rvert^{q},\lvert X-X_{n}\rvert> \varepsilon]^{\frac{p}{q}}E[\lvert X-X_{n}\rvert^{\frac{pq}{q-p}},\lvert X-X_{n}\rvert> \varepsilon]^{\frac{q-p}{q}}\\\leq \varepsilon^{p}+\left(\sup\limits_{m \in \mathbb N}E[\lvert X-X_{m}\rvert^{q}]\right)E[\lvert X-X_{n}\rvert^{\frac{pq}{q-p}},\lvert X-X_{n}\rvert> \varepsilon]^{\frac{q-p}{q}}$$
I do not know how to bound $\lvert X-X_{n}\rvert^{\frac{pq}{q-p}}1_{\{\lvert X-X_{n}\rvert> \varepsilon\}}$ that I need to prove the proposition. Any ideas?
I dont quite understand the step where you used Hölder's Inequality, i would use it as follows:
\begin{align} E|X-X_n|^p&\leq \epsilon^p+E\bigl(|X-X_n|^p\cdot \mathbb{1}_{|X-X_n|>\epsilon}\bigr)\\ &=\epsilon^p+\int |X-X_n|^p\cdot \mathbb{1}_{|X-X_n|>\epsilon}\\ &\leq \epsilon^p+\biggl(\int |X-X_n|^q\biggr)^{p/q}\cdot \biggl(\int\mathbb{1}_{|X-X_n|>\epsilon}\biggr)^{1-p/q}\\ &\leq \epsilon ^p+P(|X-X_n|>\epsilon)^{1-p/q}\bigl(\sup_{n}E|X-X_n|^q\bigr)^{p/q}\\ &=\epsilon^p+C\cdot P(|X-X_n|>\epsilon)^{1-p/q} \end{align} where $C=\bigl(\sup_{n}E|X-X_n|^q\bigr)^{p/q}<\infty$. So now, using $P(|X-X_n|>\epsilon)\to 0$ as $n\to \infty$ we get $$\limsup_{n\to \infty}E|X_n-X|^p\leq \epsilon^p$$