Today I had a very important exam about many topics and as I didn't study enough, I felt like playing chess against an opponent who was playing without a King and with 17 queens in his favor (I'm more than sure I'll get a 2/10).

I had a painful headache and couldn't even do this exercise :

Show that in the bisection method, if $\displaystyle c_j=\frac{a_j+b_j}{2}$ then $\displaystyle |c_n-c_{n+1}|=\frac{b_0-a_0}{2^{n+2}}$.

I've tried many times to solve it, but I just can't. I prefer not to show my work, but I can say I put $\displaystyle c_n-c_{n+1}=\frac{c_n}{2}$ but now that I've turned around the answer during more than 1 hour, I'm not even sure of it. I doubt about all what I know, as did Descartes one day.

I'd be happy and grateful if you could help me to solve this problem...