This should be written either in symbolic form: $\displaystyle \forall n:\text{integer}\,\neg\exists k:\text{integer}.\,n^2-2=4k$ or in words: "For all integers n, there does not exist an integer k such that $\displaystyle n^2 - 2=4k$". In the first case, the notation "n : integer", as far as I know, comes from programming languages where it means "n is a variable of type

integer". It is also possible to write $\displaystyle n\in\mathbb{Z}$ where $\displaystyle \mathbb{Z}$ is the standard notation for the set of integers. Equality should not be used to write this.

To prove this statement by contradiction, you assume that n is an integer and then assume the negation of what you have to prove, i.e., that there does exist an integer k such that $\displaystyle n^2 - 2=4k$. One way to continue is to note that in this case n^2 is even and then to use an auxiliary fact that if a square of an integer x is even (this happens only when x itself is even), then x^2 is divisible by 4. From here, a contradiction can be obtained.

Forum rules say, "9. Start a new thread for a new question. Don't tag a new question onto an existing thread. Otherwise the thread can become confusing and difficult to follow." I guess, another reason for this is that unless a new thread is opened, people don't know that there is a new question; they think that whoever answered your first question is better equipped to answer any follow-ups. Also, we love when a whole thread can be marked as [SOLVED]