How are neighbouring sequences defined? (Metric spaces)

Question

What does it mean for sequences to be neighbours in a metric space?

My attempt is:

In a metric space $(X,d)$, $(x_n)$ and $(y_n)$ are neighbouring sequences iff $$\forall_{\epsilon>0}\exists_N n\geq N \implies d(x_n,y_n)<\epsilon$$ where $n, N \in \mathbb{N}, \epsilon \in \mathbb{R}$