Application of the Well-Ordering Principle

Question

I am suppose to show that there is no infinite sequence of strictly decreasing non-negative integers and know that the Well-ordering principle will have to come into play. Also, will induction need to be involved when working through the proof? How would one begin to show this result indeed works?

Regards,

John

Answer 1

Suppose that there is a strictly decreasing sequence $(x_n)_{n\in\mathbb N}$ of non-negative ingteers. By the well-ordering principle, there is some $n\in\mathbb N$ such that$$(\forall m\in\mathbb{N}):x_n\leqslant x_m.$$Now, what can you say about $x_n$ and $x_{n+1}$?