Limit points

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Problem:

Prove that the point p is a limit point of the point set X if and only if each open point set containing p contains a poin in X which is different from p. Prove without using sequences. Only use the def. of open set, open interval, and that the point p is a limit point of the point set X means that each open interval containing p contains a point in X which is different from p.

Proof:

Definition: An open neighbourhood of a point p in a metric space (X, d) is the set V (p) = {x X | d(x, p) < }
Examples In the real line R an open neighbourhood is the open interval (p - , p + ).
Definition A subset A of a metric space X is called open in X if every point of A has an ...