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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?

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    so would it be the case that if I work with the normed spaces $X$ and $Y$ and define an inner product norm on $X\times Y$, the result still holds?2015-05-27

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  • 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.

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    If pi_x(x_n,y_n) goes to pi_x(x,y) then how can we say that x_n goes to x?similarly for y_n?2012-10-26