Maximal element, I would like a suggestion to be able to prove it

Question

Let $A$ and $B$ be partially ordered sets, and let $f:A\rightarrow B$ be strictly increasing function. Prove that if $b$ is maximal element of $B$ , then each of $f^{-1}( b)$ is a maximal element of $A$.

Answer 1

Suppose that, on the contrary, there exists some $a\in A$ such that $f^{-1}(b)<_Aa$. Since $f$ is strictly increasing, then we would have that $f(f^{-1}(b))=b<_Bf(a)$, contradicting the fact that $b$ is a maximal element in the sense of the partial order $<_B$ of $B$.