What does $(X_n, Y_n)$ mean? ($X_n, Y_n$ are two sequences of real numbers) I apologize for this very basic definition question, but I am stumped because my professor uses this notation without any introduction.
Verbatim,
"Recall that one of the properties of algebraic convergence is that $(X_n, Y_n) \rightarrow (X,Y)$ iff $X_n \rightarrow X$ and $Y_n \rightarrow Y$."
What does $(X_n, Y_n)$ mean? Google results show that $(X_n, Y_n)$ may mean an ordered pair. But then what is the definition of the convergence of an ordered pair?
 A: For a given $n \in \mathbb{N}$, $(X_n,Y_n)$ is an ordered pair. When $n$ varies, though, you get a sequence which can be denoted by $\{(X_n,Y_n)\}_{n=0}^{\infty}$.
"It is said to converge if and only if each component sequences $\{X_n\}_{n=0}^{\infty}$ and $\{Y_n\}_{n=0}^{\infty}$ converges in the usual sense."
This is actually a theorem if you define convergence in $\mathbb{R^2}$ using the distance $d:\mathbb{R}^2\to\mathbb{R}$ given by the square root of the difference of the squares of each component.
Given a sequence $\{A_n\}_{n=0}^{\infty}=\{(X_n,Y_n)\}_{n=0}^{\infty}$ with $X_n,Y_n \in \mathbb{R}$ for each $n\in \mathbb{N}$ we say it is a convergent sequence if there is an $C=(C_1,C_2) \in \mathbb{R^2}$ (a limit) and for every $\varepsilon >0$ (tolerance) there is a $N\in \mathbb{N}$ such that for every $n>N$ we have $d(C,A)<\varepsilon$
Roughly speaking, if you know a sequence converges you get a $N$, you can then use it to show that each of the component sequence has an $N_x$ and a $N_y$ that work for each of their own convergence. The other direction is similar.
