Math 2.0

Sequences, series, summability Corollary

Decreasing sequence bounded below converges

Builds on Monotone convergence theorem for sequences

Statement

Let (an)(a_n) be a decreasing real sequence: an≥an+1a_n \ge a_{n+1} for all n∈ℕn \in \mathbb{N}. Assume that (an)(a_n) is bounded below. Then (an)(a_n) converges, and its limit is the infimum inf⁡{an:n∈ℕ}\inf \{a_n : n \in \mathbb{N}\}.

Remarks

The proof applies the monotone convergence theorem to the increasing sequence (−an)(-a_n). For example, 1/n1/n decreases and is bounded below by 00. Its limit is its infimum 00. Together, the two statements say that a monotone sequence converges if and only if it is bounded.

Reading path

To understand this page from the ground up, read these 7 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 sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementDecreasing sequence bounded below converges, statement, no evidence levelDecreasing sequence b…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. Monotone sequence definition
  6. Monotone convergence theorem for sequences statement
  7. Decreasing sequence bounded below converges statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Monotone_convergence_theorem, Convergence of a monotone sequence of real numbers
About this page
Tag02V2GYAC8P
URIm2://02V2GYAC8P
Kindstatement, corollary
WritingW0 stub
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 02V2GYAC8P.
UsesMonotone convergence theorem for sequences

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