Prove that for any $a \in \mathbb{R_+}$, show that there is a number $x \in \mathbb{R}$ such that $x^2 = a$.

Question

I was looking at this proof How to prove that there is a unique positive real number $x$ such that $x^2 = 2$? today that discussed proving that there is an unique, positive real number $x$ such that $x^2 = 2$. I wanted to generalize the proof to show that for any $a \in \mathbb{R_+}$, there is a number $x \in \mathbb{R}$ such that $x^2 = a$. Any ideas?

Answer 1

Consider, $$f(x) = x^2-a $$

$$f'(x) = 2x$$

$f$ is increasing on $R_+.$

Since, $a$ is positive,

$$f(0) = -a <0$$

Again, $f(x)$ tends to infinity when $x$ tends to infinity.

Hence, $f$ has exactly one real root on $R_+.$