1. The problem statement, all variables and given/known data Show that for every a* = (a1, 1/a1), there exists another point of the form (a, 1/a) in a ball (i.e. circle, since we're in R2) of radius r, centered at a*, for any r > 0. 3. The attempt at a solution This is actually only a part of the whole problem, but I just can't put it down properly. I've tried various things on squeezing a1 < a < a1 + r, but when I invert it I can't show that 1/a < (1/a1) + r. My logic was that if I did show that, then I'd be able to show that the distance of such a point (a, 1/a) to the original one is less than r. I know just for r it wouldn't work, and I'd have to take r/2 or something, but for now I'm just trying to get to show it for r. Intuitively, I can see that this doesn't always happen for just any r, and that if a1 < a < a1 + r, it doesn't necessarily follow that 1/a < (1/a1) + r. In any case, help here would be much appreciated.