Let $r' = r - a$
I am trying to proof that neighbourhoods or open balls is an open set. I've watched this video on youtube where the guy explains how the triangle inequality comes into play. I quote
Anything that's distance less than r' from q is going to be distance less than r from p
Can someone show me how that is done? All I can write is that for some x that's between the q and r', and p and r. I get $|x - q| < r' < |x - p| < r$