mockfinal - Mock final test. 1. List all limit points,...

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Unformatted text preview: Mock final test. 1. List all limit points, supremum and infimum of the sequence (sin(πnq)) where q is a fixed rational number. What happens for an irrational q? 2. Let (xn ) be a sequence of points in a metric space X that converges to a point x. Show that it has a subsequence converging as fast as you please. In other words, if (rk ) is any sequence of decrreasing positive numbers that tends to 0, then you can choose a subsequence (xnk ) so that d(xnk , x) < rk for all k ∈ IN. 3. Suppose f is a uniformly continuous mapping of a metric space X into a metric space Y . Prove that (f (xn )) is a Cauchy sequence in Y if (xn ) is a Cauchy sequence in X. 4. Prove that the series ∞ nx (−1) n=1 2 +n n2 converges uniformly in every bounded interval, but does not converge absolutely for any value of x (x is meant to be real). 5. Show that deleting the terms 1/n for all n having digit 9 in its decimal expansion makes the harmonic series n 1/n converge. 6. Suppose f is twice-differeniable on (0, ∞), f is bounded on (0, ∞), and f (x) → 0 as x → ∞. Prove that f (x) → 0 as x → ∞. 7. Suppose α increases monotonically on [a, b], g is a continuous real-valued function and g(x) = G (x) for all x ∈ [a, b]. Prove that b b α(x)g(x)dx = G(b)α(b) − G(a)α(a) − a Gdα. a 8. Let (fn ) be a sequence of continuous functions which converges uniformly to a function f on a set E. Prove that lim fn (xn ) = f (x) n→∞ for every sequence of points xn ∈ E such that xn → x ∈ E. Is the converse of this true? ...
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