Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 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 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 30 Location : the infinite, frictionless plane of uniform density
 Subject: Re: [SOLVED]Real analysis Mon Jan 18, 2010 10:14 am  
 Very nice! Candy problem. 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 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! 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 30 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_jb_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! 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 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" 

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 Subject: Re: [SOLVED]Real analysis  
 
