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If I show this statement:

$x\in \left] a,b \right[ \Rightarrow \exists n \in \mathbb{N} : x\in \left] -\frac1{n}, 1+\frac1{n}\right[$

Have I then shown this statement:

$]a,b[ \subseteq \bigcup_{n=1}^\infty \left] -\frac1{n}, 1+\frac1{n}\right[\qquad ?$

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    What do the opposite brackets mean?2011-10-05

1 Answers 1

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Note that:

  • A set $A$ is a subset of a set $B$ if and only if $x\in A\Rightarrow x\in B$.

  • If we have $A_i$ then $x\in\bigcup A_i$ if and only if for some $i$ we have $x\in A_i$.

Combine the two result and you have indeed what you wanted.

Note, while at it, that $(-1,2)$ which is the interval for $n=1$ (since $-\frac{1}{1}=-1,\ 1+\frac{1}{1}=2$) is a superset of all the other intervals. In particular this whole union is just $(-1,2)$ to begin with.

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    @jwodder: Thanks for the correction.2011-10-04