Fixed point theorem for $f :\mathbb{R}\to \mathbb{R}$, alternative proof

Question

I would like to poof the fixed point theorem that sais that if $f:\mathbb{R}\to \mathbb{R}$ is Lipschitz continious map with lipschitz constant $L<1$ then the map has a fixed point, i.e. $\exists x \in \mathbb{R}$ such that $f(x)=x$. I don't want to consider the sequence $x_{n+1}=f(x_n)$ (I already understood this proof). I would like to do the following : take $x_0\in \mathbb{R}$. Either $f(x_0)=x_0$ and we would be done. Or without loss of generality, $f(x_0)>x_0$. Then since the slope of the curve is always lower than $1$, I would like to say that at some point $x>x_0 f(x)$ is goint to cross the slope y=x. But I can't make this formal.

Answer 1

We have $f(x) \leq L \cdot (x - x_0) + f(x_0)$ for $x > x_0$. Taking $x_1 = \max\left(\frac{f(x_0) - L x_0}{1 - L}, x_0\right) + 1$ we get $f(x_1) < x_1$.

Now, consider $g(x) = f(x) - x$. We have $g(x_0) > 0$ and $g(x_1) < 0$. As $g$ is continous, $g(x_2) = 0$ for some $x_2 \in (x_0, x_1)$.