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 [SOLVED]Real analysis

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Mohammad
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PostSubject: [SOLVED]Real analysis   Sun Jan 17, 2010 10:11 pm

Prove $\mathbb{R}^2$ has a dense subset such that no three points of it are on a straight line.


Last edited by Mohammad on Wed Jan 27, 2010 1:32 am; edited 1 time in total
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Bruno
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PostSubject: Re: [SOLVED]Real analysis   Mon Jan 18, 2010 10:14 am

Very nice! Candy problem.
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Mohammad
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PostSubject: Re: [SOLVED]Real analysis   Mon Jan 18, 2010 12:47 pm

Just to feel sweet in the morning of the first they of the week!
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Bruno
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PostSubject: Re: [SOLVED]Real analysis   Tue Jan 26, 2010 5:04 pm

Here is a solution, without the details. I worked on it with my dad so he gets half the credit!

It's easy to see that in R^n, if a sequence a_j is dense and there is another sequence b_j such that the sequence a_j-b_j converges to 0, then the sequence b_j is also dense. Left as an exercise. I love you

We construct a dense sequence of points such that the jth point is not collinear with any previous pair. Here is how we do it : take any dense sequence, say the sequence of rational points. For the jth point of this sequence take a neigbourhood of radius 1/j and pick any point which does the job inside this neighbourhood. The resulting sequence meets the requirement!
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Mohammad
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PostSubject: Re: [SOLVED]Real analysis   Wed Jan 27, 2010 1:31 am

Nice! So buy your father a pitcher of dark beer in "le dieu du ciel" Very Happy
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