Real Analysis: Proving the Greatest Lower Bound Property

[major_maths]

Homework Statement

(a) Suppose that A and B are nonempty subsets of R. Define subsets -A={-x: x$\in$A} and A+B={x+y: x$\in$A and y$\in$B}. Show that if A and B are bounded above, then the greatest lower bound of -A = - least upper bound of A and the least upper bound of (A+B) = the least upper bound of A plus the least upper bound of B.

(b) Use part (a) to prove the Greatest Lower Bound Property: Any nonempty shubset of R that is bounded below has a greatest lower bound.

Homework Equations

If 0<a and 0<b, then there is a positive integer n such that b<a+a+...+a (n summands).
If A is any nonempty subset of R that is bounded above, then there is a least upper bound for A.

The Attempt at a Solution

I've proven that first part of (a), that the greatest lower bound of -A = -least upper bound of A, but I can't figure out why the least upper bound of (A+B) would = the least upper bound of A plus the least upper bound of B. [or sup(A+B) = sup(A)+sup(B)]

 

[micromass]

Hi major_maths!

So you need to prove that

$$\sup(A+B)=\sup A+\sup B$$

Can you first prove that $\sup A+\sup B$ is an upper bound of A+B??

That is, take an arbitrary element z in A+B, can you prove that $z\leq \sup A+\sup B$??

 

[major_maths]

Thanks! I got through part (a) by proving (A+B) must be nonempty and then proving that there was an upper bound in (A+B) since both A and B had upper bounds, using the Least Upper Bound Property to prove that there must be a least upper bound since there was an upper bound to begin with.

I'm stuck again on part (b) though. I know that since inf(-A) exists, -sup(A) must exist as well. I don't know how to go about proving the Greatest Lower Bound Property from there though. I was thinking about using the Least Upper Bound Property somehow.

 

[micromass]

Thanks! I got through part (a) by proving (A+B) must be nonempty and then proving that there was an upper bound in (A+B) since both A and B had upper bounds, using the Least Upper Bound Property to prove that there must be a least upper bound since there was an upper bound to begin with.

Yes, you proved that A+B has a least upper bound. But did you prove that sup(A)+sup(B) is that exact upper bound??

I'm stuck again on part (b) though. I know that since inf(-A) exists, -sup(A) must exist as well. I don't know how to go about proving the Greatest Lower Bound Property from there though. I was thinking about using the Least Upper Bound Property somehow.

You do need to prove the least upper bound property! Just transform the inf into a sup and use the least upper bound property.