# 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.

Â© BrainMass Inc. brainmass.com May 24, 2023, 12:49 pm ad1c9bdddfhttps://brainmass.com/math/calculus-and-analysis/limit-points-2643

#### Solution Preview

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 ...

#### Solution Summary

This is a proof regarding limit points.