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For my homework, I have to prove that $[0, \infty)$ is an inductive set, and for a set to be inductive it has to contain $0$, and if $x$ is an element of $A$, then $x + 1$ must be an element of $A$. I know that $0$ is an element of a because of the bracket, but I'm not exactly sure how to word it. Do I know $0$ is an element because of the bracket (notation?) or is there some axiom that states a bracket means that the number is included in the set? Thanks!

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  • $\begingroup$ What class is this for? How much rigor and detail you need to provide probably depends on what class you're encountering this problem in. $\endgroup$ Oct 24, 2018 at 2:25
  • $\begingroup$ @CarlSchildkraut the class is called Introduction to Mathematical Reasoning. Not a lot of detail is needed, just something like "by Thm xx" or "Axiom n" $\endgroup$
    – Mikeb
    Oct 24, 2018 at 2:32
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    $\begingroup$ How is the set $[0,\infty)$ defined? $\endgroup$ Oct 24, 2018 at 2:36
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    $\begingroup$ The set $[0,\infty)$ contains $0$ by its very definition: $$[0,\infty) = \{x \in \mathbb R \mid 0 \le x\}$$ $\endgroup$
    – Lee Mosher
    Oct 24, 2018 at 2:45
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    $\begingroup$ @Mikeb No, you use the definition of $[0,\infty)$ to prove the theorem about it. Lee gives a definition, that is probably the one you’re supposed to use. $\endgroup$ Oct 24, 2018 at 3:04

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