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## 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 $\bar X$.