I want to prove that the union of some intervals forms an open cover for some segment. Any ideas on how to do this?
2026-04-03 02:57:47.1775185067
How can I prove that something is an open cover?
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If $x \in (0,1)$ then $0<x<1$. By the Archimedean property of $\mathbb{R}$, we can conclude that that there exist natural numbers $m,n$ such that $1/m < x < 1-1/n$. In particular we can take $k=\max \{ m,n \}$ to get $1/k < x < 1-1/k$. I think you can conclude your result from here.