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k3k3
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Homework Statement
Let A and B be non-empty bounded sets of real numbers.
Show the infimum of A union B is equal to the min{infA,infB}
Homework Equations
If a set is bounded below, a set called S for example, there exists a number N such that x≥N for all x in S
And if S is bounded below, then there exists a number α such that α is a lower bound of S and there is no number greater than it that is a lower bound.
The Attempt at a Solution
Let A and B be non-empty bounded sets of real numbers.
Show that the inf(A∪B) = min{infA,infB}
Since A and B are subets of A∪B, then the set, A or B, that has the smallest greatest upper bound is clearly the one that is the the result of the minimum function.
Since A union B is the entirety of both sets, the greatest upper bound will be the one from the set that contains the smallest value.
I am thinking that I need to break it up into cases when infA ≤ infB and infA > infB and some how link that to the inf(A union B).
If infA ≤ infB
Then A has the smallest greatest upper bound. When you find the infimum of A union B, it will have to be A because A is a subset of A union B.
If infA = infB then the set A=B, then A union B = A = B
If infA > infB, then use the same argument but replace A with B and vice versa.
Showing this properly is the issue I am running into. Any tips on which direction I should go in proving this?