Every finite group has a finite composition series

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., group of finite composition length)
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Statement

Every finite group $G$ has a composition series of finite length, that is, a sequence

$\{e\}◃H_1◃H_2◃\dots ◃H_k◃G$

where $K◃H$ denotes that $K$ is a proper normal subgroup of $H$, and no $K$ exists such that $H_i◃K◃H_{i+1}$ for $i=1,2,\dots ,k-1$.

That is, every finite group is a group of finite composition length.

Proof

Induction on $\left|G\right|$.

When $\left|G\right|=1$, the trivial composition series $\{e\}$ is a composition series.

Otherwise, pick a proper normal subgroup $H◃G$ with $\left|H\right|$ maximal.

Then by induction, $H$ has a composition series $\{e\}◃H_1◃H_2◃\dots ◃H_k◃H$ of finite length.

Then $\{e\}◃H_1◃H_2◃\dots ◃H_k◃H◃G$ is a composition series for $G$ of finite length.