Let $Y=(Y_n)_{n \in \mathbb N}$ be a square-integrable $\cal F$-martingale. Then $X_n:=Y_n^2$ is a submartingale with a Doob-partitioning $$X_n=M_n+A_n$$ where $M_n=(M_n)_{n \in \mathbb N}$ is a martingale and $A=(A_n)_{n \in \mathbb N}$ is previsible with $A_0=0$. We call the process $A$ the squared process of variation of $Y$ and denote it with $\langle Y \rangle=(\langle Y \rangle)_{n \in \mathbb N}$.
How can I show that $A$ is motone increasing by using the equality $$\Delta A_n=E\Bigl[(\Delta Y_n)^2 \Big \vert \cal F_{n-1} \Bigr]$$ where $\Delta A_n=A_n-A_{n-1}$ and $\Delta Y_n=Y_n - Y_{n-1}$?
$E(Z|\mathcal F) \geq 0$ almost surely if $Z \geq 0$ almost surely. Hence $\Delta A_n\geq 0$ almost surely and $A_n \geq A_{n-1}$ almost surely.