Functional Analysis: Proving Closure of Finite Sets in Metric Spaces

[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

 

[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?