Convergence of Sequence ##(p_n)## to ##p##

Bachelier
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If ##p## is a limit point of ##E## then ##\exists \ (p_n) \ s.t. (p_n) \rightarrow p##

For the sequence construction, can I just define ##(p_n)## as such:

##For \ q \in E, \ \ define \ (p_n) := \left\{\Large{\frac{d(p,q)}{n}} \right\}_{n=1}^\infty##​
 
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Are you assuming E is subset of R?

Try E = {1}.

Edit: I also think your definition of limit point is wrong.
 
pwsnafu said:
Are you assuming E is subset of R?

Try E = {1}.

Edit: I also think your definition of limit point is wrong.

Thanks. I am not looking for the def. of ##l.p.##, rather the theorem that states we can alway construct a seq. converging to any limit point.

Rudin uses a different proof, but I just thought about this one and wanted to see if it is correct.
 
Bachelier said:
Thanks. I am not looking for the def. of ##l.p.##, rather the theorem that states we can alway construct a seq. converging to any limit point.

Rudin uses a different proof, but I just thought about this one and wanted to see if it is correct.

I assume you're working in ##\mathbb{R}^n##? This is true in metric spaces. It's not true in general. Your approach is correct, though your proof is not correct as written. You need to be more rigorous about how you're selecting elements for your subsequence.
 
Notice your sequence is a collection of numbers, which are not necessarily points in your space (i.e., outside of the reals, that I can think of ). For example, in R^n, for n>1, the sequence of numbers d(p,q)/n is not a collection of points in your space.
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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