# Limit points

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 point 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.

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...od which lies completely in A.

If X=R, then we have

Definition: A set U R is called open, if for each x U there exists and > 0 such that the interval ( x - , x + ) is contained in U. Such an interval is often called an - neighborhood of x, or simply a neighborhood of x.

First we prove from left to right

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. From the definition of an open set, everything open set U containing p, there exists an > 0 such that the interval ( p - , p + ) is contained in the open set. The interval ( p - , p + ) is open interval containing p, Thus it contains a ...