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A bounded sequence that is either strictly increasing or strictly decreasing, then it must converge to some limit.

Thank you

yre
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    If exists $M$ such that for any $n \in \mathbb N$ we have that $|x_n| < M$ and (insert increasing/decreasing), then exist L such that (insert epsilon-delta formula for limit). – Mauro ALLEGRANZA Apr 13 '17 at 16:06
  • @MauroALLEGRANZA Thank you, what should be put in the insert increasing/decreasing? Am unsure of this. – yre Apr 13 '17 at 16:32

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Strictly increasing sequence:

$$(\exists M \in \mathbb{R}: \forall n \in \mathbb{N}_0: |x_n| <M)\land(\forall n \in \mathbb{N}_0: x_n < x_{n+1}) \Rightarrow \exists L \in \mathbb{R}: (\forall \epsilon>0: \exists N \in \mathbb{N}_0: \forall n > N: |x_n - L| < \epsilon)$$

Analogue for strictly decreasing sequence:

$$(\exists M \in \mathbb{R}: \forall n \in \mathbb{N}_0: |x_n| <M)\land(\forall n \in \mathbb{N}_0: x_n > x_{n+1}) \Rightarrow \exists L \in \mathbb{R}: (\forall \epsilon>0: \exists N \in \mathbb{N}_0: \forall n > N: |x_n - L| < \epsilon)$$