Every finitely generated group has simple quotient

Question

I've read that every finitely generated group has a simple quotient. Is it obvious?

Answer 1

Let $X$ be a finite set of generators of $G$.

Let $n$ be the largest number such that there is a normal subgroup $H\subsetneqq G$ containing $n$ of these generators.

The set of all such $H$ satifies the hypothesis of Zorn's lemma, and has a maximal element $H$. Now if $H \subsetneqq K\trianglelefteq G$,then $\# K\cap X > n$ and $K = G$ by definition of $n$.

$G/H$ is a simple group