Multi-dimensional sequences I was just wondering if it is possible to consider sequences in multiple dimensions? Denote $(x_{t})^{n}$ to be a sequence in dimension $n$. So the "normal" sequences we are used to are denoted by $(x_{t})^{1}$. Likewise, $(x_{t})^{2} = \left((x_{1}(t)), x_{2}(t) \right)$, etc..
It seems that for an $n$-dimensional sequence to converge, all of its component sequence must converge. Is there any utility in looking at $n$ dimensional sequences that have a "significant" number of its component sequences converge? More specifically:

Let $$(x_{t})^{n} = \left(x_{1}(t), \dots, x_{n}(t) \right)$$ be an $n$ dimensional sequence. Suppose $p$ of the component sequences converge where $p <n$. What does this tell us about the behavior of $(x_{t})^{n}$?  


Why not look at a simple example? Consider $(0,0,0),(0,0,1),(0,0,2),(0,0,3),\dots$. Two of the three component sequences converge. What would you say about the behavior of this sequence of triples? 
