Let $(x_n,y_n)$ be a sequence in the product space $X \times Y$. Prove that $(x_n,y_n)\to(x,y)$ if and only if $x_n \to x$ and $y_n \to y$.
How can we use the neighborhood idea to prove this statement? please explain me how to prove this?
Let $(x_n,y_n)$ be a sequence in the product space $X \times Y$. Prove that $(x_n,y_n)\to(x,y)$ if and only if $x_n \to x$ and $y_n \to y$.
How can we use the neighborhood idea to prove this statement? please explain me how to prove this?
Assume that $(x_n,y_n)\to (x,y)$ in $X\times Y$. Let $O_1$ and $O_2$ open sets containing respectively $x$ and $y$. Then $O_1\times O_2$ is an open set of $X\times Y$ containing $(x,y)$. By definition of convergence, we can find an integer $n$ such that $(x_n,y_n)\in O_1\times O_2$ if $n\geq N$. So for such $n$, $x_n\in O_1$ and $y_n\in O_2$.
Conversely, if $x_n\to x$ in $X$ and $y_n\to y$ in $Y$, let $O$ an open subset of $X\times Y$ containing $(x,y)$. By definition of the product topology, we can find $O_1$ open subset of $X$ and $O_2$ open subset of $Y$ containing respectively $x$ and $y$. As it's a homework question, I leave the end of the proof.