 Solution

Interpret the comparison at $N$ on $\{N<\infty \}$; values at an infinite stopping time are not required. The stopping time events make $Y_n$ adapted, and $|Y_n|\le |X_n^1|+|X_n^2|$ gives integrability. On $\{N\le n\}$ the next value is $X_{n+1}^2$. On $\{N>n\}$ it is $X_{n+1}^1$ unless $N=n+1$, and on that latter event switching only decreases the value. Consequently the pointwise inequalityholds even though $\{N=n+1\}$ need not be known at time $n$. Both indicators displayed here are ${\mathcal{F}}_n$-measurable. Taking conditional expectations and using the two supermartingale inequalities therefore givesThus $\boxed{Y\text{ is a supermartingale}.}$ This proves pasting supermartingales with downward jumps without incorrectly treating the future switching event as predictable.