Let $S$ be a semigroup. An element $a \in S$ is said to be maximal if $\langle a \rangle \subseteq \langle b \rangle \subseteq S $ implies either $\langle a \rangle =\langle b \rangle $ or $\langle b \rangle = S$.
Is it true that every infinite semigroup has a maximal element in $S$.
I have proved that every finite semigroup has a maximal element. I am unable to prove that every infinite semigroup has a maximal element.
You cannot prove it, because it is not true. Consider $S = (\mathbb Q,+)$; for any $a = \frac pq$, as long as $a \neq 0$, we have that $$ \left\langle \frac pq \right\rangle \subsetneq \left<\frac p{2q}\right> \subsetneq \mathbb Q; $$ and of course we have $\langle 0 \rangle \subsetneq \langle 1 \rangle \subsetneq \mathbb Q$.