Functional analysis

1. Jan 27, 2008

patricia-donn

Hello

I need help with an analysis proof and I was hoping someone might help me with it. The question is:

Let (X,d) be a metric space and say A is a subset of X. If x is an accumulation point of A, prove that every r-neighbourhood of x actually contains an infinite number of distinct points of A (where r>0). Using this, prove that any finite subset of X is closed.

Any help or suggestions would really be appreciated.
Thanks

2. Jan 27, 2008

jambaugh

To get started you'll need to carefully parse the definitions. Closure means all Cauchy sequences converge within the set. How will this apply to a finite set? Can't you show that there is a minimum distance among the points? How will that relate to the definition of a Cauchy sequence?

I think you're missing an assumption in the first part. Let A be a set of only one point and x be that point. Was A supposed to be a non-empty open subset?