Which definition of convergence of subsequence is correct

Question

Suppose that $(x_n)$ is a convergent sequence on a metric space $(M,d)$ with limit $x \in (M,d)$

Let $(x_{n_k})$ be the sub-sequence of the sequence $(x_n)$

Then is it more appropriate to write

1) $(x_{n_k})$ converges to $x$ if $\forall \epsilon >0, \exists N \in \mathbb{N}$ s.t. $\forall n_k \geq N, d(x_{n_k}, x) < \epsilon$

or

2) $(x_{n_k})$ converges to $x$ if $\forall \epsilon >0, \exists N \in \mathbb{N}$ s.t. $\forall k \geq N, d(x_{n_k}, x) < \epsilon$

Which one is correct and why

Answer 1

Both are correct since we assume $\{n_k\}$ is an increasing sequence of natural numbers when defining $\{x_{n_k}\}$.