In a proof I saw the use of the fact that if $X_{n}\xrightarrow{L^{2}}X$ then it follows $(X_{n})^{2}\xrightarrow{L^{1}}X^2$. Now I initially I assumed that this means an inequality of the form:
there exists $c>0$ such that $\lvert x-y\rvert ^{2}\geq c\lvert \lvert x\rvert ^2-\lvert y\rvert^2\rvert $ for $x,y\in \mathbb R\;$.
should hold. However, in a previous question on this site, it was proven incorrect.
I am particularly interested in the $L^{2},\;L^{1}$ convergence in the case of underlying probability measures, in case that makes a difference to the statement.