Math Is Fun Forum

dchilow

Member

Registered: 2007-03-05

Posts: 27

A couple more Real Analysis questions

Hi guys, thanks you were a huge help last week. I have a couple more homework questions that are troubling:

Some of these may seem very simple to you guys, but I am just getting started in this, and it is very difficult for me.

Thanks

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: A couple more Real Analysis questions

I'm going to look at 2 because it's the most interesting.

It's trickier than it looks, due to the fact that you aren't told whether a and b are rational. (If they were, you could take x = (a+b)/2, then consider the new interval (a,x) and repeat to get infinite rationals in the interval.)

I can think of a way to do it, but only because I've seen a problem on here before that helped you prove it. It went something like this:

i) Prove that for any real a and b (with a<b), there exists a natural number n such that 1/n < b-a.

ii) Now prove that there exists a natural number k such that a < k/n < b.

If you do those, you've done the first part of your question.
The irrational bit is easier, because you can use the result from the rational bit.

Consider the interval (a+π, b+π). You just proved that there are infinite rationals in that interval, and if a rational x is in that interval, then an irrational x-π will be in (a,b). Hence, (a,b) contains infinite rationals.

If you were being particularly observant, you may have noticed that the question I gave you assumed that a<b. This is a necessary assumption, and your question is being naughty for not stating it. If a ≥ b, then (a,b) is the empty set and so obviously doesn't contain infinite rationals or irrationals (because it doesn't contain anything).

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1. and 3. aren't too bad.

1. is a fairly simple proof by induction.
The fact that sup A U B = sup{supA, supB} might help you (although you might also need to prove it first).

For 3, think about what would happen if the statement was false. What could you say about B-ε in that case?

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A couple more Real Analysis questions

This is a necessary assumption, and your question is being naughty for not stating it.

Intervals are by most definitions, a < b.

1. is a fairly simple proof by induction.
The fact that sup A U B = sup{supA, supB} might help you (although you might also need to prove it first).

No need.  Maximums and minimums are defined for any finite set of numbers.  Just prove that they are as claimed.

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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..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: A couple more Real Analysis questions

One of the courses I'm on at the moment says that the a and b that define an interval don't restrict each other at all, but if a>b then:

[a,a] = {a}
(a,a) = [a,b] = Φ

So the definition must vary a bit.

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A couple more Real Analysis questions

Think of it this way: Is there really any point to allowing the interval (5, 3)?  What possible use could we have for such an interval?

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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..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: A couple more Real Analysis questions

Admittedly none, I think the definition just includes those intervals for completeness. This way, any a and b make an interval that means something.
Like I said, that's just the way I'm being taught it.

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: A couple more Real Analysis questions

Oh, I completely understand.  I'm just arguing for why I think my way is better.  Less need to worry about conditions, shorter proofs, etc.

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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..."