Is my proof of showing "4rs is even" correct?

Question

For my study I need to write proofs. I have a hard time understanding that if my answer isn't the same as in the solution book, if it then can still be correct.

The Question given is :
"Prove that $4rs$ is even , with $r$ & $s$ Integers"

Here's the proof I have written :

I assume here that a proof of $4rs$ can be written in multiple ways.

Assume that r and s are particular intergers.
It is given that the number is $4rs$.
It can be written as $4rs=2(2rs)$
Let $t= 2rs$ and note that $t$ is an integer.
Hence, there exists an integer $t$ such that $4rs=2t$
Therefore, $4rs$ is even.

This is the answer given in the solution book :

assume that $r$ and $s$ are particular integers
It is given that the number is $4rs$
It can be written as $4rs=2(2rs)$
Since r and s are integers, so are $r·s$ and $2rs$, specifically, even integers
therefore, $2(2rs)$ is an even integer
therefore, $4rs$ is an even integer

Is my proof also correct?
Any feedback is much appreciated