Let $X,Y$ be adapted, positive integrable processes and $E(X_{n+1}\vert \mathcal{F}_{n})\leq X_{n}+Y_{n}$ and $\sum Y_{n} < \infty$.
How can I define a supermartingale by the above?
Background: I want to show that $X_{n}$ converges a.s. to a finite limit, using a stopping time. It is defined as: $\tau:=\inf\{n:\sum\limits_{i=1}^{n}Y_{i}>K\}$ for some $K >0$
How can I show that $(X_{n})_{n}$ is a supermartingale from the facts $(*)$ or must I define a new supermartingale?