Bounded second moments do not upgrade convergence in probability to convergence in L2 (Bounded second moments do not upgrade convergence in probability to convergence in $L^2$)

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 $n$.