Convergence in Lp from convergence in probability and an Lr bound
= Convergence in Lp from convergence in probability and an Lr bound
If $X_n\to X$ in probability and $\sup_n\lVert X_n\rVert_r<\infty$, then $X_n\to X$ in $L^p$ for every $1\leq p<r$. The $L^r$ bound gives uniform integrability of the $p$th powers, while convergence in probability controls their bounded part.