Proving aₙ = n/(n+1) → 1

Intro to Proofs · Axiom Academy

EXAMPLE Proving a Sequence Limit Find N and write a formal – N proof that converges to 1 Let . Prove that . Concretely: given any , find a threshold N so that every term past it satisfies . Each dot is a term . They rise toward the limit line y = 1 . Fix a tolerance : the shaded strip is the band . The claim is that from some index N on, every term lands inside the strip — and shrinking just pushes N farther right. Nice work — you built a complete – N proof. Here is what carried the argument: Start from : reduce the distance to the limit to a clean closed form — here . Solve for n directly: the exact requirement , giving the tight threshold . A looser bound also works: using leads to the simpler — any sufficient N proves the limit; it need not be the smallest. The ceiling: turns a real threshold into a natural-number index. Write it in order: "Let be given," then choose N from , then verify . This is the engine behind every sequence-limit proof in analysis: bound , read off what n must satisfy, choose N , and confirm it works for all n > N .

This is the written version of the interactive lesson above. See the full Intro to Proofs course.