|May9-11, 08:39 PM||#1|
Open ball contained in another open ball
1. The problem statement, all variables and given/known data
Let (X,d) be a metric space. Suppose that a1,a2 in X and r1,r2>0 are such that the open ball B1(a1,r1) is a proper subset of the open ball B2(a2,r2).
a) Prove that r1<r2.
b)Must it always be true that r1< (3/2)*r2?
2. Relevant equations
3. The attempt at a solution
So far I have that if a2 is not in B1, then d(a1,a2)>r1, but a1 is in B2 so d(a1,a2)<r2, therefore r2>d(a1,a2)>r1.
The problem I'm having is finding some relation between r1 and r2 when a2 is contained in B1 because I end up with d(a1,a2)<r1 and d(a1,a2)<r2 which doesn't help much.
Is there a better method than using two cases like I am?
I feel like this shouldn't be difficult and that I'm just missing something small but it's driving me crazy.
Any help or hints would be greatly appreciated.
|May10-11, 12:22 PM||#2|
A problem I'm having with your proof is where you say that d(x,a2)<r1 implies x is in B1, doesn't it just imply that it is in some ball B(a2,r1), not B1? Maybe I'm missing something but it seems like something isn't correct here, can you explain?
Edit: Oh, looks like he deleted his post before I replied, can anyone else help?
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