I'm talking about [itex]E \times F[/itex], where [itex]E,F \subseteq \mathbb{R}^d[/itex]. If you know [itex]E[/itex] and [itex]F[/itex] are compact, you know they're both closed and bounded. But how do you define "boundedness" - or "closed", for that matter - for a Cartesian product of subsets of Euclidean [itex]d[/itex]-space?(adsbygoogle = window.adsbygoogle || []).push({});

The only idea I've had is viewing [itex]E\times F[/itex] as a subset of [itex]\mathbb{R}^{2d}[/itex]. If this is a legitimate thing to do, boundedness is certainly preserved. Also, since [itex]E[/itex] and [itex]F[/itex] were both closed, any sequence of points in [itex]E\times F[/itex] that converges necessarily converges to a point [itex](x,y) = (x_1,x_2,\ldots,x_d,y_1,y_2,\ldots,y_d)[/itex]. Does this look right?

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# How can you prove that a Cartesian product of compact sets is compact?

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