Math 2.0

statement theorem

Monotone convergence theorem for sequences

Statement

Let (an)(a_n) be an increasing real sequence, in the sense of a monotone sequence. Assume that (an)(a_n) is bounded above. Then (an)(a_n) converges, and its limit is the supremum sup⁡{an:n∈N}\sup \{a_n : n \in \mathbb{N}\}.

Remarks

The supremum ss exists by the completeness of R\mathbb{R}. For ε>0\varepsilon > 0 there is NN with aN>s−εa_N > s - \varepsilon, and then s−ε<an≤ss - \varepsilon < a_n \le s for all n≥Nn \ge N. The statement fails in Q\mathbb{Q}. The decimal approximations 1,1.4,1.41,1.414,…1, 1.4, 1.41, 1.414, \dots of 2\sqrt{2} increase and are bounded, but they have no limit in Q\mathbb{Q}. The node decreasing sequence bounded below converges gives the dual statement. An increasing sequence that is not bounded above tends to infinity.

Explanations

L1 human

An increasing sequence that cannot pass a ceiling must pile up somewhere. It piles up at the lowest ceiling, which is the supremum 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 …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

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Sources

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

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