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Given that $x \in R:$

$$[x,\rightarrow ) := \{y \in R\ | \ y \ge x \} = \bigcup \{[x, x + n) \ | \ n \in N\} $$

What I don't understand here is why the right equality holds (from which will follow that $[x,\rightarrow )$ is open)? Could you please point me to a proof of this fact.

Aelx
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  • Are there countably many elements of $R$ in $[x,x+n)$? – Eric Towers Feb 07 '20 at 15:41
  • The set you are looking at on the far right is indexed by a countable set but it is not ITSELF countable. The interval $[x,x+n)$ has uncountably elements, e.g. if $x=1$ and $n=1$ you are looking at $[1,2)$. – Jürgen Sukumaran Feb 07 '20 at 15:41
  • @TSF Is there a proof of this equality somewhere? – Aelx Feb 07 '20 at 16:00
  • @TSF concerning the countability, you are right $[x, x+n)$ is not countable. Right side is a union of uncountable sets. The indexing confused me. – Aelx Feb 07 '20 at 16:08
  • As for proof, by the Archimedean principle, for any $y \ge x$, there is an integer $n > y-x$, so $x \le y < x + n$, and is therefore $y$ is in the union. And every element in the union is $\ge x$, and therefore is in $[x,\to)$. – Paul Sinclair Feb 07 '20 at 23:40

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