Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$
It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof?
Theorem 2.2.1. [R. Ellis] Let $S$ be a compact right topological semigroup. Then there exists an idempotent in it.
This theorem is also known as Ellis–Numakura lemma.