I found this as a part of the proof that I was trying to accomplish. More specifically it says,
If $X_{n}$ converges to $X$ in probability i.e., $X_n \xrightarrow{P} X$ and $\mid X_{n}\mid \le Y$ $ \:\forall n\ $ and $\Bbb{E}[Y^{p}] < \infty$ then $$X_n \xrightarrow{L^p}X$$
after some calculations I came to this inequality, $$\mathbb{E}[\mid X_n-X\mid^p] < \epsilon^p+2^p\Bbb{E}[Y^p\Bbb{I}_{A_n}] \:\:\: \epsilon \ge0$$ , where $A_n :=\{\mid X_n-X\mid > \epsilon\}$
My idea is to show $\Bbb{E}[Y^p\Bbb{I}_{A_n}] \rightarrow 0$ as $n \rightarrow \infty$ , and then let , $\epsilon$ go small to prove the result. But I'm stuck at the last line i.e., I am unable to show, $\Bbb{E}[Y^p\Bbb{I}_{A_n}] \rightarrow 0$ as $n \rightarrow \infty$.
Can anyone help me prove this statement?
Thanks in Advance
By Lebesgue majorized convergence theorem you will get
$$\lim_{n\to\infty}\int_{\Omega} |X_n -X |^p dP =\int_{\Omega} \lim_{n\to\infty}|X_n -X |^p dP =0$$ bu this means that $||X-X_n ||_{L^p} \to 0.$