Difference between cluster and limit point

Question

What is the difference between cluster point of a sequence and limit point of a sequence in a metric space?

Answer 1

I think when the definitions are written out like this it is clear:

Let $(X,d)$ be a metric space and $(x_1, x_2, \ldots )$ be a sequence in $X$

A limit point of $(x_1, x_2, \ldots )$ is $z \in X$ such that if $\varepsilon \in \mathbb{R}_{> 0}$ then there exists $\ell \in \mathbb{Z}_{>0}$ such that if $n \in \mathbb{Z}_{\geq \ell}$ then $d(x_n, z) < \varepsilon$

In this case we write $z =\lim_{k \to \infty} x_k $

A cluster point of $(x_1,x_2,…)$ is $z \in X$ such that there exists a subsequence $(x_{n_1}, x_{n_2}, \ldots)$ of $(x_1,x_2,…)$ such that $z = \lim_{k \to \infty} x_{n_k}$

Looking at the definitions and noting that $(x_1, x_2, \ldots)$ is a subsequence of itself, it's true that if $z$ is a limit point of the sequence, it is also a cluster point. However, it's not always true that a cluster point is a limit point:

Consider the sequence $(0,1,0,1, \ldots)$. Both $0$ and $1$ are cluster points by the fact that $(0, 0, \ldots)$ and $(1,1,\ldots)$ are subsequences. However, neither of these are limit points with the standard metric.