What does it mean for a sequence to be monotone?

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Jase Leonard · Accepted Answer

The exact definitions are as follows:A sequence             (              x        n            )        ∈    ℝ                or              ℂ    ,    n    ∈    ℕ   is called monotone increasing       ⇔    ∃    k    ∈    ℕ   such that             x              n        +        1              ≥          x      n        ∀    n    ≥    k  .A sequence             (              x        n            )        ∈    ℝ                or              ℂ    ,    n    ∈    ℕ   is called monotone decreasing       ⇔    ∃    M    ∈    ℕ   such that             x              n        +        1              ≤          x      n        ∀    n    ≥    k  .Note also that             (              x        n            )       is said to be bounded       ⇔    ∃    M    ∈    ℕ   such that             (              x        n            )        ≤    M    ∀    n    ∈    ℕ  .In addition,             (              x        n            )       converges to a limit       x    ∈    ℝ                or              ℂ    ⇔    ∀    ε    &amp;gt;    0    ∃    N    ∈    ℕ    &amp;gt;    0   such that             |              x        n            -      x      |        &amp;lt;    ε    ∀    n    &amp;gt;    N  .Every bounded, momotonic sequence is convergent, according to another theorem.

What does it mean for a sequence to be monotone?

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