Which sequences converge in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$? Which sequences converges in the topological space $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$?

That i have tried:

For each $(p,q)\in \mathbb{N}\times\mathbb{N}$ we have that the sequence $\{(p,q),(p,q),(p,q),\cdots,(p,q)\cdots$} coverges to $(p,q)$ for all $(p,q)\in \mathbb{N}\times\mathbb{N}$.

Am I ok?

Is it correct? Do you have a different example approach?
HINT: Apart from the eventually constant sequences, note that every point which is not a successor, or the minimum, is the limit of a sequence of smaller points. Can you characterize these points? If so, then you can characterize the sequences which converge.
ALERT: sorry the next answer is not correct:

I think that you are right.

As we have that $\left(\mathbb{N}\times\mathbb{N}, \mathcal{J}_{\text{Lexicographic}}\right)$ is a discrete space, it has the trivial convergence. By definition, a space $ (X, \mathcal{T})$ has trivial convergence if the only sequences that converges are the constant sequences.