a is a cluster point of a set A iff for any $\delta > 0$, the interval $(a-\delta, a+\delta)$ contains infinitely many points of A

Question

Prove that a number a is a cluster point of a set A iff for any $\delta > 0$, the interval $(a-\delta, a+\delta)$ contains infinitely many points of A.

My work:

I found a proof for one direction in case of $\mathbb{R^2}$ in schaum series "Theory and problems of general topology" in page 56 problem no.7 but it seems to me that in this way the solution of the above question would be very long, could anyone suggest a smarter solution for me please?

Answer 1

Suppose $(a-\delta, a+\delta)$ contains only finite number of points of $A$ other than $a$, say $\{a_1,a_2,...,a_N\}$. Then there exists $\delta'$ such that $(a-\delta', a+\delta')$ contains no point of $A$ other than $a$, leading to a contradiction. Take, for example, $0<\delta' < \min \{|a_i-a|: 1\leq i \leq N\}$. The converse implication is immediate from definition.