Suppose $X^1_n, X^2_n$ are supermartingales with respect to $\mathscr{F_n}$, and N is a stopping time such that $X^1_N \geq X^2_N$. Then show $Y_n = X^1_n 1_{(N > n)} + X^2_N 1_{(N \leq n)}$ is a supermartingale and $Z_n = X^1_n 1_{(N \geq n)} + X^2_N 1_{(N < n)}$ is a supermartingale.
It seems obvious that $Y_n$ is first $X^1_n$ and once $X^1_n$ becomes greater or equal to $X^2_n$, $Y_n$ becomes $X^2_N \leq X^1_N$. Then $Y_n$ is decreasing in expectation (thus a supermartingale). I am not sure if my intuition is correct and also I am having trouble writing it down formally.
Greatly appreciate any help. I am new to rigorous probability theory and it is my first time learning Martingale.
Hint: Observe that because $X^1_N\ge X^2_N$ you have $$ Z_{n+1} \le X^1_{n+1}1_{\{N>n\}}+X^2_{n+1}1_{\{N\le n\}}. $$ Now take conditional expectations on both sides with rspect to $\mathcal F_n$, use the fact that $1_{\{N>n\}}$ is $\mathcal F_n$-measurable, and the supermartingale property of $X^1_n$ and of $X^2_n$.