Prove that $f$ has a fixed point. Let $f:[0,\infty [\to[0,\infty [$ continuous such that $$\lim_{t\to\infty }\frac{f(t)}{t}=\ell\in[0,1).$$ Prove that $f$ has a fixed point, i.e. there is an $x\geq 0$ such that $f(x)=x$. I don't really know how to solve this problem. My first intension was to use Brouwer, but it's only useable on a compact. After I tried by induction but with no success.

Consider the function $g(t)=f(t)-t$. Since $f$ is nonnegative, we must have $g(0) \ge 0$. If $g(0)=0$, we are done. Otherwise, we have $\lim_{t \to \infty} \frac{g(t)}{t}=\ell-1<0,$ so $g(t)<0$ for sufficiently large $t$. It immediately follows by the Intermediate Value Theorem that $g$ has a positive root.