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 $(x_{t})^{n}$?  

 A: You are basically taking ordered tuples over the set of sequences; absolutely no problem with that: given any set $X$, you can consider $X^n$, the set of all $n$-tuples of elements of $X$. Here, $X= \mathbb{R}^{\mathbb{N}}$, the set of all real sequences.
But just like with $n$-tuple of real numbers, knowing $k\lt n$ of the entries does not usually tell you anything about the remaining entries. It is only when you know the tuple lies within some set where there is some relation among the entries that you can hope to conclude something about the "other" entries from knowledge of some of them. 
So if you have a point $(a_1,\ldots,a_n)\in\mathbb{R}^n$, knowing something about the first $k$ entries, $k\lt n$, will not tell you anything about the remaining entries in general. But if you knew, say, that the point lies in the subset of $n$-tuples that satisfy $x_1+\cdots+x_n=0$, say, then you could deduce information about $x_n$ from knowledge about $x_1,\ldots,x_{n-1}$.
Similarly, just knowing that $p$ of the components of your tuple converge doesn't tell you anything about the remaining $n-p$ components, unless you happen to know your tuple lies in some special subset of $(\mathbb{R}^{\mathbb{N}})^n$ from which you can extract information.
A: 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? 
