## Real Analysis and Martingale Convergence in L2

Hi guys,

I am trying to prove the L2 Martingale Convergence Theorem but am struggling with some real analysis.
For a martingale {M_n} such that E[|M_n|^2] \le B < \infty,
let

A_ab = {\omega: lim inf M_n(\omega) \le a < b \le lim sup M_n(\omega)}

How is

A_ab \in {\omega: sup_{m\le k < \infty} |M_k - M_m| \ge \epsilon}

for \epsilon > (b-a)/2?

Thank you for any advice in advance.

Regards,

MancBluebird