You wish to prove that is always divisible by 2:
You know that all even numbers have a factor that is even. This is true because all even numbers can be divided by 2 given the definition of even numbers.
So now you see that you can factorize as . Then you see that and are two integers that follow each other on the real number line (like 3 and 4, 4 and 5). You can now argue that no matter what is, one of the two factors has to be an even number, since every second integer on the real number line is even.
If is an even number, then has to be an odd number. If is an odd number, then insert this expression for in . Thus
, must be an even number. Finally you know that since
always has a factor that is divisible by 2, then the expression itself must also be divisible by 2.
Mathematically, the proof looks like the one below, where the argument is constructed with implications: