A proof by induction about composition series

Question

In above lemma and proof, I can't understand how do we consider $G$ fixed. Because, I think that if $G$ fixed then, for proof's argument, if $G$ has composition series of length $n$, then it has to have all the composition series of length $1,2,\dots,n-1$; which may (and does) not hold.

This induction argument I can't digest, Please help.

(Note that I have yet to encounter Jordan-Holder, so can't use that to make my life easier.)

Answer 1

The induction hypothesis is that the statement holds

for every finite length solvable group having composition length $n$

and a solvable group $G$ having composition length $n+1$ is considered. If $$ 0=H_0\lhd H_1\lhd \dots\lhd H_n\lhd H_{n+1}=G $$ is a composition series for $G$, then $H_n$ has composition length $n$ and the statement holds by induction hypothesis.