Convergence in Lp from convergence in probability and an Lr bound

If $X_n\to X$ in probability and ${\sup }_n\Vert X_n{\Vert }_r<\infty$, then $X_n\to X$ in $L^p$ for every $1\le p<r$. The $L^r$ bound gives uniform integrability of the $p$th powers, while convergence in probability controls their bounded part.

On $([0,1],\mathcal{B},\mathrm{Leb})$, the variablesconverge in probability and in $L^1$ to zero, while $\Vert X_n{\Vert }_2=1$ for every $n$.