Proving Equality of Supremum and Infimum for Bounded Sets

[Chinnu]

Homework Statement

(a) Let A be bounded below, and define B = {b$\in$R : b is a lower bound for A}.
Show that sup(B) = inf(A).

(b) Use (a) to explain why there is no need to assert that the greatest lower bound exists as part of the Axiom of Completeness.

(c) Propose another way to use the Axiom of Completeness to prove that sets bounded below have greatest lower bounds.

Homework Equations

We can use the Axiom of Completeness, DeMorgan's Laws, etc...

The Attempt at a Solution

I have shown that both sup(B) and inf(A) exist.
I can see, logically, why they should be equal, but I can't seem to write it down clearly.

 

[micromass]

You must show two things:

1) sup(B) is a lower bound of A
2) If x is a lower bound of A, then $x\leq \sup(B)$.

Let's start with the first. How would you show that for all a in A it holds that $\sup(B)\leq a$??