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# [SOLVED]Real analysis

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Descartes

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

 Subject: [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

Posts : 184
Join date : 2009-09-15
Age : 31
Location : the infinite, frictionless plane of uniform density

 Subject: Re: [SOLVED]Real analysis   Mon Jan 18, 2010 10:14 am Very nice! Candy problem.

Descartes

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

 Subject: 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!

Posts : 184
Join date : 2009-09-15
Age : 31
Location : the infinite, frictionless plane of uniform density

 Subject: 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. 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!

Descartes

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

 Subject: 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"

 Subject: Re: [SOLVED]Real analysis

 [SOLVED]Real analysis
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