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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.

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At the fixed point $x^*=g(x^*)$ but this just means $x^*=f^{-1}(h(x^*))$ that is also $f(x^*)=h(x^*)$, that is the curves intersect at $x^*$. – Jon Jan 12 '12 at 21:34
up vote 0 down vote accepted

I will post a full answer after having seen that comments are better as a suggestion. The idea is to use the iteration technique to find the intersection between two curves $y=f(x)$ and $y=h(x)$. This is achieved by defining


assuming an inverse exists for at least one of these curves. Now, we iterate the equation


until a fixed point $x^*$ is reached, provided the criteria for such a point to exist are fulfilled. At the fixed point one has by definition


Now we turn back to our definition of $g(x)$ and we have


Inverting $f$


and so $x^*$ is the intersection point for the two curves.

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