Bounded sequence without decreasing subsequence has a minimum

Question

How to prove a bounded sequence which does not have decreasing subsequence has a minimum. I've tried to prove the contrapositive but failed. What I am wondering is that can we prove the sequence without a decreasing subsequence is increasing, if so, how to prove it, if not, then why? Thanks in advance.

Answer 1

Let $(a_{n})$ be a sequence without a decreasing subsequence and suppose $(a_{n})$ has no minimum. In particular that means that for all $n\in\mathbb{N}$ there exists an $m>n$ such that $a_{m}<a_{n}$. But this allows us to construct a decreasing subsequence.