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**Problem:**

(a) Let [tex] a_1 = a, a_2 = f(a), a_3 = f(a_2) = f(f(a)), \ldots, a_{n+1} = f(a_n),[/tex] where [tex] f [/tex] is a continuous function. If [tex] \lim _{n \to \infty} = L,[/tex] show that [tex] f(L) = L [/tex].

(b) Illustrate part (a) by taking [tex] f(x) = \cos x , a = 1,[/tex] and estimating the value of [tex] L [/tex] to five decimal places.

**My answer:**

(a) [tex] \lim _{n \to \infty} a_{n+1} = \lim _{n \to \infty} f(a_n) = f \left( \lim _{n \to \infty} a_n \right) = f(L) = L [/tex]

(b) I have used my calculator to get this one. First, I plugged in: [tex] \cos 1 [/tex]. I took the cosine of the result. Then, I kept on taking the cosine until my results agreed to five decimal places. I got [tex] L \approx 0.73908 [/tex].

**My question:**

Did I get it right? Are there other ways to find answer (b)?

Thanks a lot!!!