I was given this question as an extra question on the lecture notes. I had been thinking about this question for the past 2 weeks but to no avail. Before making this post, I have scoured through different similar posts on math stackexchange, but the questions asked are usually either
- showing that the statement is true with example or
- proof the statements in the reverse direction.
As for my thought process, as mentioned in the question, we need to prove this using proof by contradiction. So, we are going to assume that there exists some element eg Y such that (Y,X) does not appear in the relation if X is the supposed maximal element, but I have no idea on the next steps of the proof.
Sorry if I was unclear about the question as it is my first time posting here.
Let $Y$ and $X$ be according to your initial assumption, i.e. it is not the case that $Y\leq X$, where $\leq$ is the partial order. Then let $B=\{Z\in A:Y\leq Z\}$. Since $B\subset A$ is finite, $B$ has at least one maximal element $W$. Note that $X\notin B$, so $W \neq X$.
But we claim that $W$ is actually maximal for all of $A$ as well, since otherwise we would have $W< W'$ for some $W'\in A$, and then $Y\leq W< W'$, whereby $W'\in B$, contradicting the maximality of $W$ in $B$.
Since $W\neq X$ are both maximal in $A$, uniqueness is contradicted.