Math Is Fun Forum

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis Help

Hi I need some more help in Real Analysis. We are covering infimum and supremum of a set, as well as the Completeness Axiom.

Completeness Axiom: Each nonempty set of real numbers that is bounded above has a supremum.

1. The Completeness Axiom only asserts something about sets that are bounded above. Use the Completeness Axiom to prove that every nonempty set of real numbers is bounded below has an infimum.

2. Let

be a nonempty set of real numbers that is bounded above and let

. Prove that for each

there exists a point

such that

.

Where "supS" is the supremum of the set S.

Thanks

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Real Analysis Help

1.  Say you have a set S which is bounded below. Then -S = {-1*s | s ∈S} is bounded above, and so you can use the completeness axiom on that.

If -S has a supremum x, then S has an infimum -x.

2.  One of the properties of a supremum is that no upper bound can exist that is less than the supremum. You can use this in a proof by contradiction.

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Why did the vector cross the road?
It wanted to be normal.

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Real Analysis Help

Im currently taking Real Analysis as well, would appreciate if someone could check my proofs

Last edited by LuisRodg (2009-02-13 11:42:42)

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

Hey Luis, for one, you can check on some of my other posts to see where another university is at during the semester. Are you guys on a similar level at this point in the semester? It would be cool to have someone else tho chat with about this stuff. Thanks for the help everyone.

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Real Analysis Help

Hello sumpm1. In my university before we can take Real Analysis we must take a class named "Introduction to Advanced Mathematics" in which we are introduced to proofs etc. In that class, the last month or so we do an introduction to analysis in which we learn all about supremum, infimum etc.

However, this semester, in the actual Real Analysis class, in the first day we did a short introduction to complex numbers and Schartz Inequality and then we began by defining metric spaces, doing proofs about open and closed sets, compact sets, etc. We are following Rudin's Mathematical Analysis book so we will always be dealing with metric spaces instead of actually R^k. I dont know if this is good or bad. We actually do all the theorems in metric spaces instead of Euclidean space.

We already went over sequences, limits of a sequence etc and are now dealing with series.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Real Analysis Help

For #1, you should prove that -gamma is a glb.  It takes three seconds to do, but your proof won't be, uh...., complete without it.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis Help

Yes, I do need to prove that -gamma is the greatest lower bound of S, but how do I go about this?

Avon

Member

Registered: 2007-06-28

Posts: 80

Re: Real Analysis Help

The proof for question 1 that has been given is almost certainly the standard proof. However here is the idea of an another proof that I think is much prettier.

Suppose S is a non-empty set that is bounded below. Let L be the set of lower bounds for S. Show that L has a supremum and that infS = supL.