Without worrying about formal proofs for the moment decide w

Without worrying about formal proofs for the moment, decide which of the following statements about suprema and infima are true and which are false. For those that you identify to be false, give a counterexample to support your conclusion. A finite, nonempty set always contains its supremum. If a

Solution

Answer :-

(a)

True.

Explanation :- A finite non empty set always conatin a maximum .That maximum will also be the supremum

(b)

False

Explanation :- Let A=(0,1) then a<1 for all a A , but sup A=1.

(c)

False

Explanation:- Let A = (0,1) and B=(1,2) then Sup (A) = Inf (B)=1

(d)

True.

(e)

False

Explanation :- Let A = B = (0,1). Then supA = supB, but no element of B is an upper bound for A.