Math Is Fun Forum

sumpm1

Member

Registered: 2007-03-05

Posts: 42

Real Analysis Help please

Hi guys, I have a couple of bonus questions that would help me boost my grade in this class.

1. A function ƒ:R->R has a PROPER RELATIVE MAXIMUM VALUE at c if there exists d>0 such that f(x)<f(c) for all x that satisfy 0<|x-c|<d. Prove that the set of points at which ƒ has a proper relative maximum value is countable.

2. Determine whether the sequence{((-3)^n)/n!} converges and give a proof of your conclusion.

Thanks guys

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Real Analysis Help please

2. I'm not going to give a rigorous proof for this, but I will tell you that it converges.  This is easy to see if you define the sequence recursively:

As you can see, after n = 2 every successive value will be equal to the previous value multiplied by a number whose absolute value is less than 1.  The value of the sequence will bounce between positive and negative numbers, but it will converge to 0.  I leave the proof to you.

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Wrap it in bacon

Muggleton

Member

Registered: 2009-01-15

Posts: 65

Re: Real Analysis Help please

1. A function ƒ:R->R has a PROPER RELATIVE MAXIMUM VALUE at c if there exists d>0 such that f(x)<f(c) for all x that satisfy 0<|x-c|<d. Prove that the set of points at which ƒ has a proper relative maximum value is countable.

This is one of those questions which are intuitively obvious but present such a challenge to prove. I can only suggest using the metric-space properties of the real numbers. Try proving that the set of points at which ƒ has a proper relative maximum value is nowhere dense in R. Countability should follow from the fact that all nowhere-dense subsets of R are countable. (Are they? I think so anyway. )

Last edited by Muggleton (2009-02-15 02:25:35)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Real Analysis Help please

1. A function ƒ:R->R has a PROPER RELATIVE MAXIMUM VALUE at c if there exists d>0 such that f(x)<f(c) for all x that satisfy 0<|x-c|<d. Prove that the set of points at which ƒ has a proper relative maximum value is countable.

Actually, this problem isn't too bad if you're allowed to know that the real line is 2nd countable.  But a step in the right direction either way is to prove:

Let c be a proper relative maximum value.  Then there exists d > 0 such that f(x) is not a proper relative maximum value for all x in (c-d,c)U(c, c+d)

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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..."