I am attempting to self-study proof techniques and your criticism of my following proof would be greatly appreciated. Feel free to nitpick minor/trivial things; that's how I'll learn!
Edit: I have appended a revised proof with the criticisms received so that other rookies can learn from my progress too.
Edit 2: And another, third time lucky?
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We therefore have;
$$P \rightarrow Q$$
By the law of the contrapositive, we may obtain;
$$\lnot Q \rightarrow \lnot P$$
Which is translates to "If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$$n = 2m$$
By multiplying this by $3$ we may obtain;
$$3n = 6m \equiv n = 2m$$
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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Attempt 2, taking into consideration previous criticism
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We want to show that $$P \rightarrow Q$$ and that by taking contrapositives this is equivalent to showing $$\lnot Q \rightarrow \lnot P$$ which translates to;
"If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$$n = 2m$$
By multiplying this by $3$ we may obtain;
$$3n = 6m$$
This must still be even as an even integer multiplied by an odd integer produces an even one.
We must now show that $3n$ is even. If this is so, then $6m = 2k$ for some integer $k$.
As $3n=6m=2k$ we have;
$3n=2k$
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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Attempt 3
Theorem: Let $n$ be an integer. If $3n$ is odd then $n$ is odd.
Let $P$ be the sentence "$3n$ is odd" and $Q$ be the sentence "$n$ is odd." We want to show that $$P \rightarrow Q$$ and that by taking contrapositives this is equivalent to showing $$\lnot Q \rightarrow \lnot P$$ which translates to;
"If $n$ is not odd then $3n$ is not odd", or put another way; "If $n$ is even then $3n$ is even."
If an integer $n$ is even then there exists some integer $m$ such that;
$$n = 2m$$
By multiplying this by $3$ we may obtain;
$$3n = 6m$$
Which can be rewritten as;
$$3n = 2(3m)$$
Thus we have shown that $3n$ is even, as it is equal to $2(3m)$ which, as an integer multiplied by 2, must be even.
It has therefore been shown that if $n$ is even so is $3n$ and that this is equivalent to showing that if $3n$ is odd then so is $n$.
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Best Answer
The very end of the proof is not very clear. First, there is no reason to go from $3n=6m$ back to $n=2m$. Instead, in order to show that $3n$ is even, you must show that it can be written as $2k$ for some integer $k$. So what you want is to take $3n=6m$, and express $6m$ as $2k$ for some $k$. That will prove that $3n$ is even (which you haven't as yet, at least not explicitly).
I don't know what kind of course this is for; if it is for an "introduction to proofs" course, chances are you are required to be very explicit about what you are doing, so the opening of your argument is likely required and good. Later, you should be free to summarize the entire first three paragraphs of your proof into: "We proceed by contrapositive: suppose $n$ is even. Then..."
Comment on the addition: The claim that "this must be even because an even integer multiplied by an odd integer produces an even one" relies on a statement that, for all I know, has not been proven, and which is far more general than is needed here.
Simply write $3n$ as
2 x <some integer>explicitly. Given that you know that $3n$ is equal to $6m$, it should be a trivial matter to express it that way.Second: it's not the equation that "must be even". Equations (like statements) are neither even nor odd. It is integers that are even or odd (in this context); so what is the "This" it refers to? If "This" refers to $3n$, then the next line is unnecessary, but then this sentence amounts to asserting what you are trying to prove (instead of proving it).
Third: Given that you say you have to show that $3n$ is even, you cannot simply say "If this is so..." and proceed. That would mean that you are assuming what you want to prove in order to prove it, which of course is a circular argument. Of course you have concluded that $3n$ is even in that paragraph: you began by assuming it's even.
I think that you have convinced yourself that there is a clever trick or some deep thing that needs to go into showing that $3n$ is even. There isn't. There is a simple, nay, trivial algebraic manipulation that you need to do from your equation $3n=6m$ that will show that $3n$ is indeed even, and that is all you need to do.