Residually nilpotent not implies imperfect

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., residually nilpotent group) need not satisfy the second group property (i.e., imperfect group)
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Statement

It is possible for a group to be a residually nilpotent group but not be an imperfect group.

Facts used

1. Every group is a quotient of a residually nilpotent group (this in turn follows from every group is a quotient of a free group and free implies residually nilpotent)

Proof

Explicit proof

The proof is pretty direct from Step (1), but it is spelled out explicitly below.

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	Let $G$ be a nontrivial perfect group				We are using that such groups exist. For instance, we can take $G$ as a simple non-abelian group such as alternating group:A5.
2	Let $K$ be a residually nilpotent group such that $G$ is a isomorphic to a quotient group of $K$.	Fact (1)		Step (1)	Fact-direct
3	$K$ is not an imperfect group			Steps (1), (2)	An imperfect group is a group that admits no nontrivial perfect quotients. But from Steps (1) and (2), $K$ admits as quotient the nontrivial perfect group $G$. Thus, $K$ is not imperfect.
4	$K$ is residually nilpotent and not imperfect.			Steps (2), (3)	Step-combination direct

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