I'm trying to prove that $\sum_{i=1}^n|x_i| \geq |\sum_{i=1}^nx_i| \forall \bar{x} \in \mathbb{R}^n$. Intuitively, I know this is true. I also can provide counter examples to say that $\sum_{i=1}^n|x_i|$ cannot always be less than $|\sum_{i=1}^nx_i|$, and the two terms are not always equal to one another, but I don't know how to prove the $\geq$ relationship.
Are disproving the other two possiblities enough to prove a $\geq$ relationship? If not, can anyone help me figure out where to start proving it?