Well-order on infinite set with property

Question

The problem. I want to prove the following theorem:

Let $X$ be an infinite set. Then, there is a well-order $\leq$ on X with the property: $$|\{ y \in X : x < y\}| = |X|, \ \forall x \in X.$$

My approach

By Zorn's lemma, there is a well-order on this set. Now I've tried to get to a contradiction by looking at the set $\{x : |\{y \in X : x < y\}| < |X|\}$: this set has a least element because $X$ is a w.o., thus the entire set has cardinality smaller than $|X|$. Now here I'm stuck. I'm not sure if I'm taking a correct approach, or if there's another way to prove this.

Answer 1

Let k = |X|. As the set of all ordinals with cardinality k,
is not empty, there is a least ordinal $\beta$, called an
initial ordinal, with k = $|\beta|.$

Map $\beta$ onto X.