Suppose that f : R->R is conitnuous and that its image f(R) is bounded. Prove that there is a solution for the equation f(x)=x for x in R
Let $\displaystyle m=\inf f(\mathbf{R}), \ M=\sup f(\mathbf{R})\Rightarrow m\leq f(x)\leq M, \ \forall x\in\mathbf{R}$
Let $\displaystyle g:\mathbf{R}\to\mathbf{R}, \ g(x)=f(x)-x$
$\displaystyle g(m)=f(m)-m\geq 0, \ g(M)=f(M)-M\leq 0$
g continuous $\displaystyle \Rightarrow \exists c\in[m,M]$ such as $\displaystyle g(c)=0\Rightarrow f(c)=c$