How does one write this in quantifiers?

Question

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

Thank you

Answer 1

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:

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