Hello,

I need help with the following problem:

Let S = {(x,y) in R2| xy>1}. Show that S is open.

This is what I've done so far:

1. Let u be a point in S.

2. So, u = (u1,u2) for some u1, u2 in R and u1*u2>1

3. I chose my epsilon to be equal to (u1*u2 - 1) / ||u||, which is > 0.

4. Now I need to show that the open ball B(u, epsilon) is contained in S.

So, let v = (v1,v2) be a point in the ball B(u, epsilon).

That means dist(u,v) < epsilon.

In order for v to be in S, we need to show v1*v2 > 1. However, I'm having difficulty showing this. I tried to use the inequality dist(u,v) < epsilon, but I didn't get anything about v1*v2. Any suggestions on what should I do? Appreciate anyone's help and thanks for reading my post