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sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis help please

Hi. I have some homework problems in a Real Analysis class that I am taking. These are a couple of problems that I am having trouble with. This stuff is very difficult to understand. Thanks in advance.

Last edited by sumpm1 (2008-02-12 14:52:53)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis help please

You just need to get the hang of the language of the mathematical symbols. Once you’ve managed it, it shouldn’t be so difficult.

1(a)
Let x ∈ (a,b). Then

Let

Then

i.e.

Hence (a,b) is open.

1(b)
Let

Then either x < a or x > b.

In both cases,

Hence

is open – i.e.

is closed.

2
The middle inequality comes straight from the definition of infimum and supremum and the fact that S is nonempty: since it is nonempty, there is an element x in S; then, by defintion, infS ≤ x ≤ supS.

To prove infA ≤ infS, assume to the contrary that infS < infA. Then infA is not a lower bound of S (since infS is the greatest of the lower bounds of S), and so ∃ s ∈ S with s < infA. This would be a contradiction since S is a subset of A and so s ∈ S ⇒ s ∈ A ⇒ infA ≤ s.

The proof for supS ≤ supA is similar.

Last edited by JaneFairfax (2008-02-12 23:40:38)

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Re: Real Analysis help please

Hi Jane, I thoroughly appreciate your help. Seeing how you broke things down to their definitions helps alot. But there seem to be tricks that you need along the way. I am still having a little trouble justifying step 3 in problem 1a though:  I am not exactly sure what property I can pin this on. Thank you so much. I am glad there is someone out there that has actually seen this stuff before!

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Real Analysis help please