Is what you did correct?

f_n+1(x) - f_n(x) is decreasing (I believe). This should imply that (m_n) is decreasing. And it is bounded by [-1,0] so it must be convergent.

As n goes to infinity, f_n(x) goes to 1/(1-x) on (-1,0]. How do I find the limit A though (to what (m_n) converges)?

Thanks for your help.