Let a and b be non-empty, bounded subsets of r. (a) why does sup(aub) exist? (b) prove that sup(aub) = max{sup a, sup b}.

Answer with Step-by-step explanation:

Let A and B be non- empty bounded subset of R

a.We have to find why exist

If A and B are bounded set

Then there exist  constant  such that

and

Then , sup of A =b and sup of B=d

When  a set is bounded then all elements lie in the set are lie between the constants s and t.

All elements are less than or equal to t then t is supremum of set.

Because both set are bounded and sup of both set A and B are exist.All elements A union B are less than or equal to sup A or sup B.

Then, exist.

b.We have to prove that

Suppose ,A =(1,2) and B=(2,3)

Sup A=2 , sup B=3

Upper bound of

Hence,

If A=(4,5),B=(2,3)

Sup A=5,Sup B=3

Hence,

Hence, we can say that .

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step-by-step explanation:

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