Let A be a nonempty bounded set and suppose that S is anonempty subset of A. Pro

Question

Let A be a nonempty bounded set and suppose that S is anonempty subset of A. Prove that inf A inf S sup S sup A. Some definitions: Supremum = sup: Suppose that S ( a nonemptyset of real numbers ) is bounded above. A number is thesupremum of S if is an upper bound of S and any number lessthan isnot an upper bound of S. We will write = supS. Infimum = inf: Suppose that S ( same as above) is bounded below. A number is the infimum of S if is a lower bound of S and any number greater than is not alower bound of S. We will write = inf S. Let A be a nonempty bounded set and suppose that S is anonempty subset of A. Prove that inf A inf S sup S sup A. Some definitions: Supremum = sup: Suppose that S ( a nonemptyset of real numbers ) is bounded above. A number is thesupremum of S if is an upper bound of S and any number lessthan isnot an upper bound of S. We will write = supS. Infimum = inf: Suppose that S ( same as above) is bounded below. A number is the infimum of S if is a lower bound of S and any number greater than is not alower bound of S. We will write = inf S.

Explanation / Answer

Since each s in S is also in A we have that inf A

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