Can A be a limit point if there exists an epsilon neighborhood

[kathrynag]

Homework Statement

Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)\capA={a}

Homework Equations

The Attempt at a Solution

A point is an isolated point if it is not a limit point.
Let a be an element of A.
Let be an isolated point. We want to show V(a)\capA={a}.
Since a is not a limit point, we say x=lima_{n} satisfying a_{n}=x

 

[jbunniii]

Try proving this equivalent statement:

A is a limit point iff there does not exist an epsilon neighborhood V(a) such that V(a) \cap A = \{a\}.

Notice that the following mean the same thing:

"there does not exist an epsilon neighborhood V(a) such that V(a) \cap A = \{a\}"

"every epsilon neighborhood V(a) contains at least one point of A that is distinct from a."