Thread: Real Analysis Question - Proving Supremum

Hello. I am new to this forum and this is my first question.

Let A be bounded below, and define B = {b is an element of R : b is a lower bound for A}. Show that sup B = inf A.

Any help on this would be greatly appreciated. Thanks.

Take the set B, which is bounded above, since for b \in B, and a \in A, a > b, so all a are upperbounds of b. Call sup B = c. Then for any d< c, d is not an upperbound of B, so d is not in A. So c <= a \in A, and c \in B. Then for any e st e>c, then e is not in B, since c is the supremum of B. Then c is also the inf A, so inf A = sup B

I just have one small additional question. Throughout the proof you use the notation a\ in A. I was unsure as to what the \ denoted. Thanks again.

nothing really.