I have the following task to do:

Let $(X\_n)$ and $(Y\_n)$ be two supermartingales on the probability space $(\Omega, \mathcal{A}, P)$ and $T$ be a stopping time regarding a filtration $(\mathcal{F}\_n)$ and $X\_T \leq Y\_T$ on $\{T< \infty\}$.

Define $Z\_n = Y\_n$ on $\{n<T\}$ and $Z\_n = X\_n$ on $\{T \leq n\}$.

Proof that $(Z\_n)$ is a supermartingale.

I tried to use $Z\_n = Y\_nI\_{\{n<T\}} + X\_nI\_{\{T\leq n\}}$ and plug that into $\mathbb{E}[Z\_n|\mathcal{F\_{n-1}}]$ and use some properties of the conditional expectation, but I don't seem to get to $\leq Z\_{n-1}$.

Any help is appreciated.