Prove Let $A$ and $B$ be ordered classes,and let $f:A\to B$ be an increasing function. Prove that if $a$ is the greatest element of $A$, then $f(a)$ is the greatest element of $f[A]$
Attempted Proof
By definition, $ x \leq a $ for every $x\in A$
Then $f[A]=\{f(x)|x\in A\}$ Since $f$ is increasing then $ x \leq a \to f(x) \leq f(a) $
Thus $f(a)$ is the greatest element of $A$.
I feel something is missing. Any help would be appreciated. I thought of doing it by contradiction