Math 2.0

Sequences, series, summability Sequences and limits Proposition

A sequence converges iff its liminf and limsup agree

Builds on Limit inferiorLimit superiorConvergent sequenceSqueeze theorem

Statement

Let (an)(a_n) be a bounded real sequence. Then (an)(a_n) converges if and only if lim inf⁡n→∞an=lim sup⁡n→∞an\liminf_{n \to \infty} a_n = \limsup_{n \to \infty} a_n. In this case, lim⁡n→∞an\lim_{n \to \infty} a_n is equal to this common value.

Remarks

The proof of the direction from right to left uses inf⁡k≥nak≤an≤sup⁡k≥nak\inf_{k \ge n} a_k \le a_n \le \sup_{k \ge n} a_k and the squeeze theorem. In the extended real line, the statement holds for every sequence. Then convergence to +∞+\infty or to −∞-\infty is allowed. For (−1)n(-1)^n, the two values are −1-1 and 11. They differ, so the sequence diverges.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionLimit inferior, definition, no evidence levelLimit inferiordefinitionLimit superior, definition, no evidence levelLimit superiordefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementA sequence converges iff its liminf and limsup agree, statement, no evidence levelA sequence converges …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Limit inferior definition
  6. Limit superior definition
  7. Convergent sequence definition
  8. Squeeze theorem statement
  9. A sequence converges iff its liminf and limsup agree statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior, Properties
About this page
Tag0HM732ZNVC
URIm2://0HM732ZNVC
Kindstatement, proposition
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0HM732ZNVC.
UsesLimit inferior, Limit superior, Convergent sequence, Squeeze theorem

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