I was reading a theorem about martingale in a textbook by Yuan Shih Chow.However,I cannot understand a detail in the proof,and I will appreciate if you could explain me in the red frame why we can choose $n_1$ large enough to ensure $P\{V_0 - A_1\}<\delta /4$. I think this is impossible since $A_1$ and $V_0$ cannot be close enough. Moreover,I also considered of it might be a typo that $C_n^0$ maybe $B_0\{S_n>1,S_j\leq 0,for m_0\leq j< n\}$.But that also seems incorrect. That is the picture of the textbook,where in the red frame I can't understand
2026-03-29 03:36:34.1774755394
A detail in the proof of a martingale theorem about a.s. convergence under the stopping time condition
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