- #1
Visceral
- 59
- 0
Homework Statement
Suppose [itex]x,y \in X[/itex] which is a normed linear space and [itex]x\neq y[/itex]
. Prove that [itex]\exists r>0[/itex] such that [itex]B(x,r) \cap B(y,r)=∅[/itex]
Homework Equations
Epsilon Ball
[itex]B(x,r)={z \in X:||x-z||<r}[/itex]
The Attempt at a Solution
So my attempt here is via contradiction and its not workout out. I assume the intersection of the balls around x and y are not empty, and try to use the fact that x is not equal to y and use some typical triangle inequality tricks to attempt to get a contradiction. I always end up getting what I assumed to be true.
For instance, if the intersection is not empty then there exists an element in both balls. This would imply that the 2*r>ε=||x-y||. However I can't contradict this. What am I doing wrong? Please give any suggestions if you can. I missed this problem on a test and I still can't figure it out.