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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