Engineer ends 10-year struggle with 上極限 by asking AI

The post is framed as a personal study note rather than a research result. The author, an engineer by trade who began studying probability and machine learning roughly a decade ago, says 上極限 was one of the concepts that stayed out of reach the longest. The turning point came not from a textbook but from AI chat: the author notes that being able to ask relentless follow-up questions, without the social friction of a tutor or a manager, is what made the material stick. The explanation itself is a step-by-step reconciliation of two definitions. In set terms, the inner union from k=n onward shrinks as n grows, and taking the intersection across all n captures events that occur infinitely often — the author's own paraphrase is that no matter how far you go, it happens again. In sequence terms, the supremum from n onward is monotonically decreasing, so by monotone convergence it has a limit, which is the 上極限. The bridge between them, per the author, is the same operation in both cases: keep shifting the starting point forward, discard anything that happened only finitely often early on, and keep what remains at the end. The author closes by predicting that learning itself will change in a fundamental way as a result of this kind of access — a hedged personal view, not a claim backed by data.