Consider two curves $y=f(x)$ and $y=h(x)$, both are one-to-one (invertible). Define $g(x)= f^{-1}(h(x))$ (inverse of $h(x)$). Assuming the scheme $X_{n+1}= g(X_n)$ converges to a fixed point $x^*$, show that at $x^*$ the curves $f$ and $h$ intersect.
I have tried graphing various functions to see how I can prove this but to get started I don't really even get what to do here... I think this is fixed point iteration but I am not good a proving proofs so I need some help getting started.