Infinity in multidimensional space

Question

Let's say we have a sequence $x_n \in \mathbb{R}^2$ such that $\lim_{n \rightarrow \infty} {x_n} = \infty$. What is infinity here? In other words, if $x_n \rightarrow (a, b)\,(n \rightarrow \infty)$ and $(a,b) = \infty$, then does this mean that $(a,b) = (\infty, \infty)$ or does it mean that at least $a$ or $b$ is equal to $\infty$?

Answer 1

if it was, say: $$|x_n|\to\infty$$ this would make sense, but having a 2D value tend to a 1D value is just ambiguous, it could mean any of the following: $$\begin{pmatrix}\infty\\\infty\end{pmatrix},\begin{pmatrix}\infty\\\ne\infty\end{pmatrix}$$ etc