3.10 Let $f\in\mathcal{C}^1[-1,1]$ such that $|f(t)|\leq 1$ and $|f'(t)|\leq\frac{1}{2}$ for all $t\in[-1,1]$. Let $A=\{t\in[-1,1]\colon f(t)=t\}.$ Is $A$ nonempty? If the answer is 'yes', what is its cardinality?
Well, I was trying like suppose $\exists t_1 \ni f(t_1)=t_1$ then if I apply mean value theorem on $[t_1,x]$ then $f(x)-f(t_1)=f'(c)(x-t_1)$ $|f(x)-t_1|\le \frac{1}{2}|x-t_1|$ then I can not proceed, where I can use the fact $|f(t)|\le 1$?