Let A and B be nonempty bounded sets. Prove that Is there an analogous result fo

Question

Let A and B be nonempty bounded sets. Prove that Is there an analogous result for A B, assuming thatthis set is nonempty? Provide either a proof or acounterexample.

Explanation / Answer

If a is the inf of A and b is the inf of B then for any x in Aunion B x is in A or x is in B. If x is in A then x > or =inf A and if x is in B then x > or = inf B. Therefore forany x in A union B x > or = min {inf A, inf B} = y. Such anumber y where for any x in A union B x > or = y is the inf of Aunion B. The same argument can be used for sup by switching theinequality. inf ( [-1,1] intersect [0,1] ) = 0. This gives motivation toa possible proof that inf(A intersect B) = max {inf A, inf B} andsup(A intersect B) = min {sup A, sup B}. See if you can provethese. If you can't then I will help.

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