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Order 9 of the following sentences so that they form a logical proof by contradiction of the statement:

If the square of an integer is odd then the original integer is also odd.

Choose from these sentences:

$n^2$ is even
Suppose $n$ is even.
$\exists k \in \mathbb{Z}$ such that $n = 2k+1$ .
Suppose $n$ is odd.
$\exists j \in \mathbb{Z}$ such that $n^2 = 2j$
Let $n \in \mathbb{Z}$ such that $n^2$ is odd.
Therefore, $n$ is odd
$n^2 = 4k^2$
So $n^2$ is odd and $n^2$ is even.
$\exists k \in \mathbb{Z}$ such that $n = 2k$ .
Suppose $n$ is an integer such that $n^2$ is even.
Presume that the given statement is incorrect.

Your Proof:

Choose from these sentences:

$n^2$ is even
Suppose $n$ is even.
$\exists k \in \mathbb{Z}$ such that $n = 2k+1$ .
Suppose $n$ is odd.
$\exists j \in \mathbb{Z}$ such that $n^2 = 2j$
Let $n \in \mathbb{Z}$ such that $n^2$ is odd.
Therefore, $n$ is odd
$n^2 = 4k^2$
So $n^2$ is odd and $n^2$ is even.
$\exists k \in \mathbb{Z}$ such that $n = 2k$ .
Suppose $n$ is an integer such that $n^2$ is even.
Presume that the given statement is incorrect.

Your Proof: