$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

Question

From Williams' Probability with Martingales

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How is the remark deduced from the proof of $b$? I really don't see it.

Answer 1

The partial sums of $\sum X_k$ are bounded by constant $c$ iff $|M_r|\le c$ for every $r$. This last occurs iff $T_c=\infty$ (where I write $T_c$ instead of $T$ to make the dependence on $c$ explicit). Therefore $$ \{\text{partial sums of $\sum X_k$ are bounded}\} = \bigcup_{c=1}^\infty \{T_c=\infty\}.\tag1 $$ So if the LHS has positive probability then $P(T_c=\infty)>0$ for at least one $c$ (else the LHS has prob zero). Now continue with the rest of the proof of (b).

The Remark shows that we can relax the requirement that $\sum X_k$ converges a.s. to the stated assumption on the partial sums.