Proving existence of numbers with intermediate value theorem

Question

How do you use the intermediate value theorem to prove the existence of numbers? For example, with $f(x) = c^2 = 2$, how can I prove that $\sqrt2$ or a positive number $"c"$ such that $f(x)$ is true exists?

I'm guessing that through the definition of the intermediate value theorem, which defines intervals of continuity, I'll have to guess with intervals around my desired value. Is that the right way to approach this?

Answer 1

We don't prove that numbers exist. We assume them. That is, we construct a model which we work in, and that model is very often the real numbers. Usually, we construct the real numbers using Dedekind cuts. Using that definition, it is fairly obvious that the square root of $2$ exists.

To prove that the square root exists, you can also use the intermediate value theorem to prove that there exists such a real number $x$ that $x^2=2$, of course. But using that theorem already assumes that you have real numbers to start with.

Answer 2

Consider the function $$g(x)=x^2-2$$ $g(x)$ is continuous since it is a polynomial. Note that $g(1)=-1<0<2=g(2)$. Hence, for some $c\in[1,2]$, we must have $g(c)=0$. But then $c^2-2=0$ so $c^2=2$.