The Epsilon-N Definition

Real Analysis · Axiom Academy

LESSON The Epsilon-N Definition When does ? Exactly when every tolerance is met from some index N on: . Take the sequence , whose terms march toward L=0 . Draw a horizontal band of half-width around L — the set of all values within of the limit. Watch the dots: the early ones land above the band, but from some index N on, every term falls inside and never leaves. Here . A term is inside the band exactly when , i.e. when n>5 . So the first term inside is n=6 , and N=6 works : for all we have . 2. Tighter Tolerance, Later Cutoff Convergence cannot rely on one lucky — it must hold for every , however small. Watch the band tighten in stages. Each time shrinks, the band narrows and the cutoff N slides further to the right : a stricter tolerance forces you deeper into the sequence before the tail is trapped. For with L=0 , solving gives , so the smallest cutoff is (the first integer strictly past ). The band-and-cutoff pairs from the animation: Putting it together gives the formal definition. Think of it as a game: a challenger names any (the band appears), and you must respond with an N that works (the cutoff snaps in and the whole tail turns green inside the band). The sequence converges precisely when you can win this game for every challenge. It must work for every positive tolerance, not just convenient ones. This is what pins the terms to L . For each you supply some cutoff N . It may depend on ; it need not be optimal.

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