Levy's Upward Theorem in Mean Square

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(\mathcal{G}_n)_{n\in\mathbb{N}}$ be a sequence of sub-$\sigma$-algebras of $\mathcal{F}$ indexed by the natural numbers that is nondecreasing: $\mathcal{G}_n\subseteq\mathcal{G}_{n+1}$ for every $n\in\mathbb{N}$. Let

be the generated $\sigma$-algebra of the union, which is again a sub-$\sigma$-algebra of $\mathcal{F}$ because $\mathcal{F}$ is a $\sigma$-algebra containing every $\mathcal{G}_n$. Let $X$ be a square-integrable random variable on $(\Omega,\mathcal{F},P)$.

Then for every choice of conditional expectations $Y_n$ of $X$ given $\mathcal{G}_n$ ($n\in\mathbb{N}$) and every conditional expectation $Y_\infty$ of $X$ given $\mathcal{G}_\infty$, the real sequence of mean-square distances $(\lVert Y_n-Y_\infty\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$. In the notation of Conditional Expectation of a Square-Integrable Random Variable: