Series of nested normal subgroup (composition series) induces a sequence of quotient groups

Question

In Group theory in a nutshell by A. Zee on pg. 66 he introduces sequences of nested normal subgroups:

$G \rhd H_1 \rhd H_2 \rhd \dots \rhd H_k \rhd I$.

Then he says that this induces a sequence of quotient groups:

$G/ H_1 \supset H_1/H_2 \supset H_2/H_3 \supset \dots \supset H_k$.

Why is this true? And how to interpret this sequence of quotient groups? At first sight it seems strange, as $G/H_1$ consists of cosets of $H_1$ and $H_1/H_2$ consists of cosets of $H_2$, it seems $H_1/H_2$ cannot be a subset of $G/H_1$.

Edit There is an errata available for this statement, it says $H_k$ in the second line should be replaced by $H_{k-1}/H_k$. This doesn't change my question.

Answer 1

It is not true, as pointed out in a comment. What is probably meant is that

this induces a sequence of quotient groups: $G/H_1$, $H_1/H_2$, $H_2/H_3$, $\ldots$, $H_{k-1}/H_k$, $H_k/I$.

Here $H_k/I \cong H_k$, so writing

$G/H_1$, $H_1/H_2$, $H_2/H_3$, $\ldots$, $H_{k-1}/H_k$, $H_k$

or

$G/H_1$, $H_1/H_2$, $H_2/H_3$, $\ldots$, $H_k$

would also be fine.