I'm asked to prove that every open ball is both open an open and closed set
So far, I've managed to show it's open:
Given a ball $ B=B(x,r) $
I made a new ball, $ B=B(y,\delta)$
let z be an element of $ B(y,\delta) $ and showed that $ d(z,x)\leq d(z,y)+d(x,y)< \delta+d(x,y)= r $
Concluding that $ B(y,\delta) $ is contained in $B(x,r)$ and therefore $B(x,r)$ is an open set Is this right?
As for showing that that the open ball is a closed set, I'm at a loss.