- #1
bobbarker
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Homework Statement
Find sup{[tex]\epsilon[/tex]| N_{[tex]\epsilon[/tex]}(X_{0} [tex]\subset[/tex] S} for
X_{0} = (1,2,-1,3); S = open 4-ball of radius 7 about (0,3,-2,2).
Homework Equations
If X_{1} is in S_{r}(X_{0}) and
|X - X_{1}| < [tex]\epsilon[/tex] = r - |X - X_{0}|
then X is in S_{r}(X_{0})
The Attempt at a Solution
This is my first foray into n-dimensional analysis and I'm pretty intimidated. I understand the theory behind it I think--we're making a little n-dimensional ball around the point X_{0}, and in fact trying to maximize the size of the ball while still remaining in the bigger ball our set is defined by.
I'm confused about the actual determination of the size of [tex]\epsilon[/tex] though. What do I use for this arbitrary point X in the equation?
This is a multi-part problem and I could do the ones in dimensions I could visualize. :P Can anyone help me out with this 4D stuff? Thanks.
Edit: the answer in the back of the book says the supremum is 5.