Math 2.0

statement corollary

Decreasing sequence bounded below converges

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.

Explanations

L1 human

A decreasing sequence that cannot pass a floor must settle down. It settles at the highest floor, which is the infimum of its terms.

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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
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Monotone_convergence_theorem, Convergence of a monotone sequence of real numbers

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