thanks in advance 3 Suppose that A is bounded above B subset

thanks in advance!

3. Suppose that A is bounded above, B subset A and B is non-empty. Prove that sup B exists and sup B

Solution

3)given A is bounded above set hence A has supremum,And B be the subset of A,implies B is bounded above set hence B also have supremum

suppose A=(-inf,p) and p be supremum of A

and  B=(-inf,q) and q be supremum of B

since B is subset of A .p=>q,equality holds if A=Bhence

sup B<=supB

4)

supremum is (5)^1/3 ,A={x:x belongs to Q, x³< 5}

The set is non empty since 0 definetly belongs to A

A is bounded above hence A has supremum