My textbook Analysis I by Amann/Escher defines the notions of neighborhood and cluster point as follows:
Then the authors present equivalent definitions of cluster point:
I would like to ask if the below definition of cluster point is equivalent to the one given in my textbook?
$a \in X$ is a cluster point of $(x_{n})$ if, for each $U \in \mathcal{U}(a)$, there is some $m \in \mathbb N$ such that $x_m \in U$ and $x_m \neq a$.
Thank you so much for your help!
No, take $x_n=a$ for all $n\in \mathbb{N}$. Then, any neighbourhood of $a$ contains every element of the sequence, and so, in particular, infinitely many terms. As such, $a$ is a cluster point of $(x_n)_{n\in \mathbb{N}}$.
In general, we want it to be true that if $(x_n)_{n\in \mathbb{N}}$ is convergent, then it has a unique cluster point: The limit.