Two different but equivalent definitions of cluster point

Question

I've seen two different definitions of Cluster point of a sequence:

Given a sequence $\mathop{\left\{x_{n}\right\}}\limits_{n∈ℕ}^{}$ of a topological space $X$, then $x∈X$ is a cluster point (accumulation point) of the sequence if for every neighborhood $V⊆X$ of $x$ there exist infinitely many $n∈ℕ$ such that $x_n∈V$

The other one is as follows:

Given a sequence $\mathop{\left\{x_{n}\right\}}\limits_{n∈ℕ}^{}$ of a topological space $X$, then $x∈X$ is a cluster point (accumulation point) of the sequence if for every $n_{0}∈ℕ$ there is some $n\ge n_{0}$ such that $x_n∈V$

The second definition is actually claiming no matter how large $n_0$ is, we always are able to find a term $x_n$ of the sequence which is close enough to the point $x$

but I cannot proof the equivalency between these two definitions

Answer 1

Let $x$ be a clusterpoint according to the first definition and let $n_0\in\mathbb N$. If there is no $n\geq n_0$ such that $x_n\in V$ then the set $\{n\in\mathbb N\mid x_n\in V\}$ is a finite set which is not the case. So we conclude that such $n$ exists. Proved is now that $x$ is a clusterpoint according to the second definition.

Let $x$ be a clusterpoint according to the second definition. If the set $\{n\in\mathbb N\mid x_n\in V\}$ is finite then it has a greatest element $n_1$. If we take $n_0=n_1+1$ then for every $n\geq n_0$ we have $x_n\notin V$ which is not the case. So we conclude that $\{n\in\mathbb N\mid x_n\in V\}$ is not finite. Proved is now that $x$ is a clusterpoint according to the first definition.

Answer 2

This doesn't really involve any topology, it's just a more or less obvious fact about sets of natural numbers:

Define $$A=\{n\in\Bbb N:x_n\in V\}.$$The equivalence of the two definitions follows from

Exercise. The set $A\subset\Bbb N$ is infinite if and only if for every $n_0\in\Bbb N$ there exists $n>n_0$ with $n\in A$.