a) Let Xn satisfy the recursion X(n+2) = 3-X(n-1)-X(n-2). ( *Note : This reads:

Question

a) Let Xn satisfy the recursion X(n+2) = 3-X(n-1)-X(n-2). (*Note: This reads: X sub n-2 equals 3 minus X sub n-1 minus X sub n-2)

(1) If Xn converges then show that its limit is 3

(2) Let Xn be defined by x1=4, x2=2, and for all n "greater than or equal to" 1, X(n+2) = 3-X(n+1)-Xn (*Note: This reads: X sub n+2 = 3 minus X sub n+1 minus X sub n)

b) Let f: R--->R be continuous and g: R--->R be any bounded function. If a is "within" R and f(a) = 0 then prove that h(x) = f(x)*g(x) is continuous at a. (*Note: R is the real numbers)

c) Let f: R--->R be continous. Suppose also that lim f(x) "as x--->+infinity" = lim f(x) "as x---> -infinity" = 0 while f(0) = 1. Prove that f has an absolute maximum on R. (*Note: R is the real numbers)

Explanation / Answer

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