I'm having trouble figuring out how can I show that if two sets are compact then their cartesian product is also compact.
Any help is much appreciated,thank you!
2026-04-03 04:38:02.1775191082
Compactness and cartesian product
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HINT: There is an $r\in\Bbb R$ such that $X\cup Y\subseteq[-r,r]$; why? What can you say about $X\times Y$ and $[-r,r]^2$? For closedness, think about the set $(\Bbb R\setminus X)\times\Bbb R$, among others.
(As noted in the comments, you can use just about any characterization of compactness to prove this result without enormous difficulty; I’ve simply chosen the one that you already had in mind.)