Suppose that is a continuous, bounded, strictly increasing function.
(a) Show that there is a point such that
(b) For each , define . Explain why . Then explain why for all
(c) Explain why the sequence is bounded above.
(d) Explain why the sequence converges to some number, L.
(e) Explain why f(L) = L.
My work so far:
(a) f is bounded and strictly increasing, therefore f'(x) > 0, so there is a point such that x < f(x). I know this reason is not enough, still working on the proper proof.
(b) Now as defined by part (a).
let , then
Now , thus proved
Therefore S is inductive, and proved
(c) Because f is continuous and bounded.
(d) Because f is monotonic, implies is monotonic, thus is bounded. All bounded monotonic sequence converges, thus the sequence converges to L.
(e) converges to L, then converges to L because f is continuous.
I know I made some mistakes up there, please check, thank you.