elementary number theory - How to approach this modulo proof?

I'm really stuck on how I might begin approaching the following proof; I've tried a few things, which I've listed at the bottom, and my own inklings of what I might try, but I'm thoroughly stumped. Here's the question I'm trying to answer"

$$\forall\: m \in \mathbb{Z}, m^2 \mod{7} = {0,1,2,4}$$

I've tried breaking this into cases, where m is either odd or even, and then trying to find the remainder for m alone and using the fact that
$$a \equiv b \mod{d}$$ and $$c \equiv e \mod{d}$$

then $$ac \equiv be \mod{d}$$

And just using this to square the results. I've also tried going back to the definition of modulo, but I can't solve the floor function I get:

$$m = 2k - 7(floor(\frac{2k}{7}))$$

Can anyone help me out here? Really struggling to figure out how to prove this :S