# Show that sup(B)=inf(A)

1. Sep 24, 2011

### Chinnu

1. The problem statement, all variables and given/known data

(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.

2. Relevant equations

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

3. 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 cant seem to write it down clearly.

2. Sep 24, 2011

### micromass

Staff Emeritus
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 $\sup(B)\leq a$??