Basically, I tried proving that multiplying an odd and even number together gives you an even number to a friend of mine. This is what I said.
Let's first take a random number $k$. It doesn't matter if its odd or even. If we then multiply this by $2$, we get an even number (from the definition of a even number, which is basically a number that can be divided by $2$). Now we know that in the number line, it goes odd, even, odd, even,..., and so if we add $1$ to this, then we get an odd number. We now have an odd number and an even number and so lets multiply them.
$$2k \cdot (2k + 1) = 2k(2k + 1) = 4k^2 + 2k) = 2(2k^2 + k)$$
We now have a random number $(2k^2 + k)$ multiplied by $2$ and so this is an even number (by definition of an even number).
My friend then said that we make the assumption that $2k \pm 1$ is odd because the number line alternates between odd and even numbers, how do know that this is true?
I was thinking how to prove this. Would it be by some form of induction?
EDIT: I think I need to change "number" to "integer" in this post don't I?