7:41 AM
I have recently bumped this question. (To make the title more descriptive.)
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Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in N$ , $g(x_n) = f(x_{n+1})$. Show that there exists $x_0 \in [a,b]$ such that $g(x_0) = f(x_0)$. Can someone provide an example of f...

Do you think tag would fit there? It does not have exactly the form of looking for a fixed point; but it seems a bit related to me. (It has the same "feel".)