There is no rush for this. I am reading my analysis book and this is left without a proof... so if anybody wants to offer help, that would be great.

Prop: If A is a subset of a metric space X, then for a point x which is an element of X, TFAE:

(i) x is an accumulation point of A

(ii) For all ε>0, Uε(x)∩A is infinite.

(iii) For all ε>0, (Uε(x)\{x}) ∩A is infinite