MATH 210 Final, Fall 2010

Math 210 - Final (Fall 2010) T. Tlas 1. (15 points) Prove that if K C R is compact and f : R — R is continuous then f(’) is compact as well. 2. (7 points each) Consider the following: i- Prove that the series 7° 9 sini) converges for every x. ii- Let f(z) = 7? sinfns) Prove that f(a) is continuous. iii- Prove that hs f(z) dx =0. 3. (15 points) Suppose f(x) is a real-valued continuous function on [0,1]. Assume that Se a” f(x)dx=0, n=0,1,2,.... Prove that f(z) =0 Vz € (0, 1]. 4. (15 points) Suppose {a;, }°2, is a sequence in R and suppose L is a real number. Assume that every subsequence of {an}92, has a sub-subsequence which converges to L. Prove that {an}%2y converges to L. 5. (17 points) Prove that <sin(x) < a for all x € (0, 3]. i 6. (17 points) Suppose g(x) is Riemann integrable on {a, b] and g(x) > 0. Assume f(r) is continuous on [a,b]. Prove that there exists ac € [a,b] such that P f(x) g(x) dx = f(c) yp g(x) dx. Mn