Sequences Summary

Real Analysis · Axiom Academy

How convergence captures "approaching a limit," and how completeness — every Cauchy sequence converges — sets apart from . Convergence is the precise promise: given any tolerance, the tail eventually stays that close to L . The limit laws, the squeeze theorem, and monotone convergence let you find limits without returning to the definition each time. Every bounded sequence has a convergent subsequence (Bolzano–Weierstrass) — and / pin down its extreme behavior. The Cauchy criterion proves convergence without naming the limit ; it is equivalent to convergence precisely because is complete . Completeness — equivalently the least-upper-bound property — is what makes limits, continuity, and integration possible in . A sequence converges to L when, for every tolerance , all terms past some index N lie within of L . "Eventually arbitrarily close" made exact. When to use: proving a limit from scratch, or that one doesn't exist. Watch out for: N may depend on , but never on the running index n . A sequence fails to converge in distinct ways: oscillation (terms never settle, e.g. (-1)^n ), divergence to (terms grow without bound), or genuinely erratic behavior with no single limit. When to use: classifying the way a limit fails. Watch out for: " " is a divergence pattern, not a finite limit. Core Concept Algebra of Limits

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