- #1
skriabin
- 11
- 0
Homework Statement
Prove that the set D = {z in C; |z^2 - 1| < 1} is open
Homework Equations
The Attempt at a Solution
I have to show that for any z in D, there exists r > 0 s.t. the nbhd N(z,r) is contained in D. Let w in N(z,r) => |z - w| < r. Need to show |w^2 - 1| < 1 for some r > 0.
I tried |w^2 - 1| = |(w^2 - z^2) + (z^2 - 1)| <= |w - z||w + z| + |z^2 - 1|,
but since we have |z^2 - 1| < 1 already, it doesn't seem to help much. Not sure what else to try. Thanks for any help!