MathDB
Closed set

Source: IMC 2002 day 1 problem 4

October 7, 2005
functionreal analysisreal analysis solved

Problem Statement

Let f:[a,b]→[a,b]f : [a, b] \rightarrow [a, b] be a continuous function and let p∈[a,b]p \in [a, b]. Define p0=pp_0 = p and pn+1=f(pn)p_{n+1} = f(p_n) for n=0,1,2,...n = 0, 1, 2,.... Suppose that the set Tp={pn:n=0,1,2,...}T_p = \{p_n : n = 0, 1, 2,...\} is closed, i.e., if x∉Tpx \not\in T_p then ∃δ>0\exists \delta > 0 such that for all x′∈Tpx' \in T_p we have ∣x′−x∣≥δ|x'-x|\ge\delta. Show that TpT_p has finitely many elements.
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