Choosing a $x \in$ $X$, with $X$ an infinte set, $<$ a well-ordering on $X$, such that $x < x'$ for only finitely many $x' \in X$

Question

Let $X$ be an (countable or not) infinite set, $<$ a well-ordering on this set.

Lately I read a proof in which was explicitly stated to choose an element $x \in X$ such that there are infinitely many $x' \in X$ satisfying $x<x'$. Clearly $x_0$ the minimal element of $X$, which exists because of the well-ordering, would do the job. But wouldn't any element of X do the job?

I cannot think of any example where $X$ is an infinite set and any x that does not satisfy this condition. Is there such an example?

Answer 1

No, if the well order has a maximal element, using it is a counterexample.