# How to prove that 0 is an element of $[0, \infty)$?

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!

• 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. Oct 24, 2018 at 2:25
• @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" Oct 24, 2018 at 2:32
• How is the set $[0,\infty)$ defined? Oct 24, 2018 at 2:36
• The set $[0,\infty)$ contains $0$ by its very definition: $$[0,\infty) = \{x \in \mathbb R \mid 0 \le x\}$$ Oct 24, 2018 at 2:45
• @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. Oct 24, 2018 at 3:04