Question

Prove the following:

Suppose that is nonempty and bounded below. Then exists.

Answer 1

Solution of Let B THEOREM Let A be a non empty subset of R, bounded below. Then A has an infimum Proof Let lo be lo be a lower bound fee I EIR: l is a lower bound of AY . Then B is non empty subset of IR because loEB also, XEB and a E A = x sa This shows that B is bounded above B is a non empty subset of R, bounded above, By the supremum Property of IR , B has supremum Let Sup B = L Then (i) 6CL for every beB since L is an upper bound of B Thus

and (ii) Since every a E A is an an upbest bound of B and L=supB is L<a fost every a EA (ii) shows that L is a Lower bound of A and (i) shows that L> any Lower bound of A. Consequently L= infa A has an infimum. The Proof is complete.