We need to show that the Denjoy homeomorphism constructed may actually be made $C_1$.
a)For each integer $n$,let $$l_n=\frac{1}{(|n|+1)((|n|+2)}.$$Show that $$\sum_{n=-\infty}^{\infty} l_n <\infty$$$$\lim_{|n|\to\infty} \frac{l_{n+1}}{l_n}=1.$$
b)For each $n$,let $I_n=[a_n,b_n]$ be an interval with length $l_n$. Define a map $f$ on $[a_n,b_n]$ by $$f(x)=a_{n+1}+\int_{a_n}^{x} 1+\frac{6(l_{n+1}-l_n)}{l^3_n} (b_n-t)(t-a_n) dt.$$Show that $f'(a_n)=f'(b_n)=1.$
c)Prove that $f$ takes $[a_n,b_n]$ onto $[a_{n+1},b_{n+1}]$ in one-to-one fashion.
I already solved a and b part and can anyone give me any hints on the c ones?