Martingale with bounded increments converge?

Question

Prove or disprove the following arguments:

1. There exists a martingale $\left(M_{n}\right)_{n}$ with bounded increments such that $\lim _{n \rightarrow \infty} M_{n}=\infty$

2. There exists a martingale $\left(M_{n}\right)_{n}$ with bounded increments such that $\limsup _{n \rightarrow \infty} M_{n}=\infty$

I am not sure how to to go about dis-proving statement 2 but for statement 1, I was going to use Martingale Convergence Theorem as $M_n$ is uniformly integrable martingale but I am not sure I am heading in the right direction.