Prove: a point a is a cluster point of a set $A \subset \mathbb R$ iff there exists a sequence ${a ^{(k)}} \subset A\setminus\{a\}$ converging to $a$.
My thoughts:
I know that the definition of a cluster point $a$ of a set $A \subset \mathbb R$ is, for every $\delta > 0$, the n-ball $B_{\delta}(a)$ contains at least one point of A, not counting $a$. but I do not know how to use this definition to prove what required.
Could anyone show me how to prove this please?
Best Answer
Hint:
Take $\epsilon = \frac{1}{n}$ in the definition of a cluster point to find a sequence. For the other direction, the fact that a sequence in $A \setminus \{a\}$ exists already tells you something about the intersection of open balls with $A$.