Sequence Convergence

Intro to Proofs · Axiom Academy

LESSON The ε–N Definition of Convergence What it really means for a sequence to "approach" a limit — and how to prove it by answering every ε with a matching index N. 1. The Challenge: a Band Around L Convergence is a game between two players. An adversary names a tolerance ε > 0 , drawing a horizontal band of half-width around the limit L . To win, we must produce an index N such that every term from a_N onward lands inside that band — and stays there forever. "For every challenge ε, there is a response N…" "…past which every term is within ε of L." 2. A Worked Proof: Solving for N(ε) Claim: the sequence converges to 0 . To prove it we don't guess N — we solve the band condition for n , and read off the smallest index that works. is equivalent to — an explicit threshold. Take , the first integer strictly past . Every clears the threshold, so the whole tail sits inside the band. We want a strict inequality . When is a whole number — say , so — the ceiling gives N = 10 and , only equal . Adding one ( N = 11 ) pushes strictly inside: . Either choice proves convergence; keeps the inequality strict. The same band picture forces a structural fact: a convergent sequence has exactly one limit. If the tail must eventually live inside every band around L , it cannot also live inside every band around a different value M — there isn't room for the tail to be in two separated places at once. Proof (contradiction). Suppose , and set the tolerance to a third of the gap:

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