When is a fact 'obvious enough' that it does not need proving? 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?
 A: 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.
A: 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.
