I'm tasked with figuring out if the following statement is correct.
$$ \forall r \in \mathbb{R}, \exists s \in \mathbb{R},s^2 =r $$
Right now my understanding is that the set of real numbers includes negative numbers.
- Assume r is a negative real number $-1$
$$r=-1$$ $$s^2 = -1$$
- Take the root of both sides
$$s = \sqrt{-1} = i$$
- Since s can be $i$, an imaginary number for some real number $r$ the statement is false
$$ i \notin \mathbb{R} \rightarrow s \notin \mathbb{R} $$
And thus by contradiction it's false. My question is whether or not this proof is valid and rigorous enough. Is it safe to assume that there's no other value that s can take on that might make the statement true?
