= Integrable overshoot of a stopped L1-bounded martingale
{title2=$\mathbb E\bigl[|f_\tau|\mathbf1_{\{\tau<\infty\}}\bigr]\leq\sup_n\mathbb E|f_n|$}
For an integrable discrete-time <martingale>, stop at the first time $|f_n|\geq N$. For any deterministic $m$, conditioning the terminal value $f_m$ on each event $\{\tau=j\}$ gives $\mathbb E|f_{\tau\wedge m}|\leq\mathbb E|f_m|$. <Monotone convergence theorem> therefore proves the displayed overshoot bound. The maximum of the <stopped martingale> is at most $N\vee |f_\tau|\mathbf1_{\{\tau<\infty\}}$, an integrable variable. This permits a convergence proof without assuming bounded overshoots.
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