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Here is an example to better explain the question:

Theorem: If $n$ is any integer, then $3n^3 + n + 5$ is odd

Counterexample:

$n = 2k + 1$

$3n^3 + n + 5 = 3(2k + 1)^3 + (2k + 1) + 5$

= odd + even + $5$

= even

Must I prove that for any value of $k$, $3(2k+1)^3$ will return an odd number? Is it enough to tell the reader that this returns an odd function, as I did, or must I show them?

Is it obvious enough that odd + even + $5$ return an odd number? Must I prove that in more detail too?

I estimate that it is up to the mathematicians discretion to decide when enough has been explained, but where about is that line drawn in practice?

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sammy
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    There is no counterexample, the assertion is true. And when you give a "counterexample" make it concrete, a specific number. – André Nicolas Jul 08 '16 at 01:15
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    As far as the base question of "do I really need to prove this every time" if it is your first class on proof writing, I would. Especially if some of the focus is on integers and their properties such as parity like this question. Once you get further, though, it is more than forgivable to omit such details. In similar fashion, in calc 2 you might take the time to prove $2^n\gg n^3$ for sufficiently large $n$, but in later courses it is not worth the effort. Write with your audience in mind. – JMoravitz Jul 08 '16 at 01:25
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    Totally depends on context and your intended readers. If your readers are at an elementary level and you judge that they will need a proof that the cube of an odd number is odd, then you should provide one. If, to go to the other extreme, you were writing a paper for a scholarly journal, there is no way you would spend time proving that. – David Jul 08 '16 at 01:28
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    Your counterexample is no longer a counterexample if you correct the key error: the middle term $(2k+1)$ is odd, not even. –  Jul 08 '16 at 01:59
  • If you provide a counterexample, it's better to provide a specific counterexample, not a class of counterexamples. There's no reason to use the variable $k$ here. If you try any specific example, you will see that it is not a counterexample and that the statement is true. – Jair Taylor Jul 08 '16 at 02:01

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In practice, it partly depends on the intended audience. Still, as a rule of thumb, one should always prove it to oneself, first, so that one can actually make sure the result is as trivial as it seems to be. In the case of your example (as has been pointed out in the comments), you were trying to prove in your "counterexample" that if $n$ is odd, then $3n^3+n+5$ is even. This turns out to be false (as you should have seen if you'd worked it out in full detail). If you'd completed a proof of something that were true, then you'd want to determine which steps of your proof you could safely assume your intended audience would follow implicitly, and omit them.

Cameron Buie
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I think that stating some results (lemmas or theorems) as an "enough clear" one completely depends on whom you are writing to (in a book) or talking to (in a lecture). Sometimes, you may see in an advance math book, the writer state some hardly understandable result as a clear one and does not provide any proof for it while there may exist some proof for it in another book.

Majid
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