Proving that a set of relating elements is infinite

Question

I want to prove the following:

Let $<$ be a well-founded relation on a set X such that $\leq$ is a total order. Show that the set $$\{x\in X: x < y\}$$ is infinite for some $y \in X$.

I really have no clue on how to go about this. I tried using the definition of well-founded relations and total orders but got nowhere...

Any help is appreciated, thanks in advance!

Answer 1

Let $X = \{ 1, 2 \} $ with the usual order. The set $X$ is well-founded and $\leq$ is a total order on $X$. Notice that $\{ x \in X \colon x < y \} $ is finite for all $y \in X$.

If this is a problem from some text, perhaps something was omitted in your description of the problem.

Answer 2

It is not true, e.g. natural numbers with natural order.