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**Induction** [FONT="]Question: Prove than an n ∈ ℤ is even if and only if n + 6 is even. Show the if part and the only if part.

My attempt...

__Only if part__

Prove that an integer n is even only if n + 6 is even.

Assume n is even; n = 2k.

Now, n + 6 is equivalent to 2k + 6.

2(k + 3) where k + 3 is obviously an integer.

[/FONT]∴ Integer n is even only if n + 6 i[FONT="]s even.

__If part__

Prove that if n + 6 is even; n is an even integer.

Assume n + 6 is even; n + 6 = 2k.

n = 2k - 6.

n = 2(k - 3) where k - 3 is obviously an integer.

[/FONT]∴ If n + 6 is even; integer n is even.

Have I made any mistakes? I have properly read the "if" and "only if" parts out of the question? I want to be sure as I'm new to proofs.