How can I prove this? Should I take $x$ and $x+2$ or not ? I am confused.
Any even number has the form $2n$. (Why? No matter what you make $n$ to be, $2n$ will, be divisible by $2$.).
Any odd number has the form $2n+1$. (Why? Play with this by plugging numbers into $n$.).
So, add two odd numbers:
Is your result always divisible by $2$? Why or why not?
Would you be able to reproduce the above with understanding?
Suppose there is greatest even integer N Then For every even integer n, N ≥ n. Now suppose M = N + 2. Then, M is an even integer. [Because it is a sum of even integers.] Also, M > N [since M = N + 2]. Therefore, M is an integer that is greater than the greatest integer. This contradicts the supposition that N ≥ n for every even integer n. [Hence, the supposition is false and the statement is true.]
Let m and n be odd integers. Then, m and n can be expressed as 2r + 1 and 2s + 1 respectively, where r and s are integers. This only means that any odd number can be written as the sum of some even integer and one.
when substituting lets have m + n = (2r + 1) + 2s + 1 = 2r + 2s + 2.