Homework 2, Real Analysis

Due August 31

Problem 1

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

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


2. $latex \displaystyle \int_a^b f_n \to \int_a^b f$.

Problem 2

Consider the functions

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

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

Problem 3

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

1. There exists $latex C>0$ such that $latex |a_n| \le \dfrac{C}{|n|^k}$.


2. The series $latex \sum a_n e^{inx}$ converges uniformly if $latex k\ge 2$.

Problem 4

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