Homework 2, Real Analysis

Due August 31

Problem 1

Let f_n:[a,b]\to\R be Riemann-integrable and f_n\rightrightarrows f.

  1. f is Riemann-integrable on [a,b]. (Hint: Given \e > 0, find a partition \mathscr P of [a,b] such that U(f,\mathscr P) - L(f,\mathscr P) < \e, where U(f,\mathscr P), L(f,\mathscr P) are the upper and lower sums of with respect to \mathscr P, respectively.)
  2. \displaystyle \int_a^b f_n \to \int_a^b f.

Problem 2

Consider the functions

\displaystyle f_n(x) = \frac{x}{1 + nx^2}.

Then f_n\rightrightarrows 0, but f_n'(0)\not\to 0.

Problem 3

Let f\in C^k(\mathbb S), a k-continuously differentiable periodic function, with period 2\pi, and let a_n be its nth Fourier coefficient.

  1. There exists C>0 such that |a_n| \le \dfrac{C}{|n|^k}.
  2. The series \sum a_n e^{inx} converges uniformly if k\ge 2.

Problem 4

Let X be a metric space and \bar X its completion. Then, for each x\in\bar X, there exists a sequence x_n in X such that x_n\to x in \bar X.

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