Definition of a cluster point

Question

A point $a\in \big< X,d \big> \quad$ is a cluster point of a sequence $(x_n) \quad $ if $$ \forall \epsilon \gt 0, \forall n\in \mathbb N , \exists p\gt n \implies d(x_p, a)\lt \epsilon $$

Can I interpret this definition as :

A point $a\in \big< X,d \big> \quad$ is a cluster point of a sequence $(x_n) \quad $ if $$ \forall \epsilon \gt 0, \exists N \in \mathbb N , n\geq N \implies d(x_n, a)\lt \epsilon (*) $$ since they look so alike, and I cannot figure out the difference between them.

$(*)$ is actually how we define a convergent sequence in a metric space. Since $\forall \epsilon \gt 0$ $B(a,\epsilon )$ contains all but finitely many terms of $(x_n)$ where $a$ is the limit of $(x_n)$

Answer 1

At the second definition $(x_n)\to a$ but the first one says that there is a sub-sequence of $(x_n)$ like $(x_{a_i})$ such $x_{a_i}\to a$.

Answer 2

No, you cannot. In $\mathbb R$, with the usual distance, take $x_n=(-1)^n$. Then $1$ is a cluster point of this sequence according to the usual definition, but not according to the alternative version that you suggested. If you take $\varepsilon=1$, then there is no $N\in\mathbb N$ such that$$n\geqslant N\implies|1-x_n|<1=\varepsilon.$$