Math Is Fun Forum

dchilow

Member

Registered: 2007-03-05

Posts: 27

Bonus questions help for Real Analysis.

1)  Prove that every decimal expansion represents a real number.

2)  Let A and B be nonempty sets that are both bounded above.  Define a set C by C={a+b:a∈A,b∈B}.  Prove that C is bounded above and supC=supA+supB. 

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Prove that the supremum and infimum of the open interval (a,b) are b and a, respectively.

Don't even know how to start. 

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Prove that the set S={x∈R: (x^2)-25x>0} is unbounded.

I know that a set S is unbounded if for each number M there is a point x∈S such that |x|>M.

I just don't know what to do from here.  I am not sure if I am supposed to prove by contradiction or directly.  I tried to do it by contradiction, but I don't know how to set it up.  Please help me on this, thanks.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Bonus questions help for Real Analysis.

dchilow, if you have multiple related questions, please post them in the same topic.

1. Think about what a decimal representation really means.  You should be able to construct sequence of rational numbers which approaches your "decimal representation" which is Cauchy.

2. If you know what a diameter is, use it.  If not, it may be more natural to assume that sup C > sup A + sup B, and then assume sup C < sup A + sup B, reaching a contradiction in each case.  Just remember, "sup" means within arbitrary distance (epsilon).

3. This one should be really straightforward.  First show that there is no number less than a and greater than b in (a, b).  Now show that any number in (a, b) falls in between a and b.  Writing out (a, b) = {x in R : a < x < b} may help.

4.  Factor it.  x^2 - 25x = x(x - 25).  When is this greater than 0?

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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..."