OK, here are two questions out of Nathanon's *Additive Number Theory* from the section on fractional parts ($\S$4.4). 

I think I'm missing something. I don't understand what there is to prove? Let $x\in \mathbb{R}$ and let

$$||x||:= \min(|n-x|:n \in \mathbb{Z}) $$ 

So, its just the distance from $x$ to the nearest integer. Questions 1 and 2 of $\S4.4$ are to show:

$$||\alpha +\beta|| \le ||\alpha||+||\beta||$$
and this little thing:
$$||x|| = ||-x|| = ||x+n||, \forall n \in \mathbb{Z}$$
I've seen similar questions before on a course in ele number theory, and didn't do so well on them. As stated above, I don't know *what there is to prove*, or how I would do it. Do I actually need to go back to the definition of $||x||$? Thanks for any help on this..