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Rudin 2.7

Rudin 2.7

av Leo Lundberg -
Antal svar: 3

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Why are we using t = min here and not t = max? If we take the union of the two sets then wouldnt we want to choose the largest radius to make sure we stay outside of the border for the limit point when taking the union? I would have understood it if it was an intersection instead

Som svar till Leo Lundberg

Sv: Rudin 2.7

av Oliver Lindström -
You are right in that we want to choose a radius such that we “stay outside the border” but if we were to let t = max(r,s) then this need not be the case. If t= r>s for example then we know there are no x \in E such that 0s. This is why we need to choose a t that is smaller than or equal to both r and s. Does this make sense?
Som svar till Oliver Lindström

Sv: Rudin 2.7

av Leo Lundberg -
Yes I think I understand now, I think I got a bad mental visualisation of the problem and was mistaken with not realising that the two points should be completely disjoint from each set E and F and the neighborhood according to the radius should also be completely disjoint from both sets, so a smaller radius is clearly more desireable in case the other radius starts reaching into the other set. Thank you for the help!