Given $\{X_i\}$ independent nonnegative r.v.'s such that for all $n ≥ 1$ we have that $E[X_n] = 1,\forall n$, define $M_n:=\prod\limits_{k=1}^{n} X_k$ and the filtration $\mathcal F_n=\sigma(X_1,\dots, X_n)$
If $\prod\limits_{k=1}^{\infty} E(\sqrt{M_k})>0$ then show that $\{S_n\}_n$ is a Cauchy sequence in $L^2$, where $S_k:=\sqrt{M_k}$
So I need $E\left((S_n-S_m)^2\right)<\epsilon$ for certain $N\le m\le n$ large enough
Set $a_k:=E(\sqrt{X_k})$ then if $\prod\limits_{k=1}^{\infty}E(\sqrt{M_k})>0\iff$ $\sum\limits_{k=1}^{\infty}(1-a_k)<\infty$
So $a_k\uparrow 1$ with growing $k$, then
$E\left((S_n-S_m)^2\right)=E\left(M_n+M_m-2S_mS_n\right)=2E(M_m)-2E(M_m\prod\limits_{k=m+1}^{n}\sqrt{X_k})$
$=2E(M_m)-2E(M_m)\prod\limits_{k=m+1}^{n}E(\sqrt{X_k})$
and this not going to lead anywhere ?