Question 10

Let $\mathbb{X} = \{ x_n: n \in \mathbb{N}\}$ be a sequence of positive real numbers, such that $\mathbb{X}$ has no least element.
Prove that $\mathbb{X}$ has a monotonic convergent subsequence.
Also give an example, with proof, of such a sequence $\mathbb{X}$, that has at least two monotonic convergent subsequences, whose limits are different.