These feel like basic enough questions, but I don't know where to find the answer.

Let $X_1,X_2,X_3,\dots$ be a supermartingale such that $|X_{n+1} - X_n| < K$ for all $n$ ($K$ fixed). Does the event $X_n \rightarrow +\infty$ necessarily have probability zero? What if we also have the condition that given $X_1,\dots,X_n$ there are only $N$ possibilities for $X_{n+1}$, each with probability $1/N$?

Suppose in addition that $\mathbb{E}(X_{n+1}|X_1,\dots,X_n) \le X_n - c_n$, where $\sum c_n \rightarrow +\infty$. Is $X_n \rightarrow -\infty$ almost certain?