= Solution
The conclusion printed as convergence in probability is already an assumption; the substantive conclusion is convergence in $L^p$, and we prove that stronger statement.
First, the almost-sure subsequence result from part (a) and Fatou's lemma show that $X\in L^r$:
$$
\mathbb E|X|^r
\leq\liminf_k\mathbb E|X_{n_k}|^r<\infty.
$$
Consequently
$$
\sup_n\mathbb E|X_n-X|^r<\infty.
$$
Fix $1\leq p<r$ and $\delta>0$. Splitting according to
$|X_n-X|\leq\delta$ and applying Hölder's inequality on the complement gives
$$
\begin{aligned}
\mathbb E|X_n-X|^p
&\leq\delta^p+
\mathbb E\!\left[|X_n-X|^p
1_{\{|X_n-X|>\delta\}}\right]\\
&\leq\delta^p+
\big(\mathbb E|X_n-X|^r\big)^{p/r}
\mathbb P(|X_n-X|>\delta)^{1-p/r}.
\end{aligned}
$$
The second term tends to zero because of convergence in probability and the uniform $L^r$ bound. Taking the upper limit and then $\delta\downarrow0$ proves
$$
\boxed{\|X_n-X\|_p\longrightarrow0}.
$$
This is <convergence in Lp from convergence in probability and an Lr bound>.
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