# Fall 2019

## 1. 
Let $\{a_n\}_{n=1}^\infty$ be a sequence of real numbers.

a. Prove that if $\displaystyle\lim_{n\to∞} a_n = 0$, then $\displaystyle\lim_{n\to∞} a_1 + \cdots + a_n = 0$.
$$
\lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0
$$

b. Prove that if $\displaystyle\sum_{n=1}^{\infty} \frac{a_{n}}{n}$ converges, then 
$$
\lim _{n \rightarrow \infty} \frac{a_{1}+\cdots+a_{n}}{n}=0
$$

## 2. 

Prove that
$$
\left|\frac{d^{n}}{d x^{n}} \frac{\sin x}{x}\right| \leq \frac{1}{n}
$$

for all $x \neq 0$ and positive integers $n$.

> Hint: Consider $\displaystyle\int_0^1 \cos(tx) dt$

## 3. 
Let $(X, \mathcal B, \mu)$ be a measure space with $\mu(X) = 1$ and $\{B_n\}_{n=1}^\infty$ be a sequence of $\mathcal B$-measurable subsets of $X$, and
$$
B \definedas \theset{x\in X \suchthat x\in B_n \text{ for infinitely many } n}.
$$

a. Argue that $B$ is also a $\mathcal{B} \dash$measurable subset of $X$.

b. Prove that if $\sum_{n=1}^\infty \mu(B_n) < \infty$ then $\mu(B)= 0$.

c. Prove that if  $\sum_{n=1}^\infty \mu(B_n) = \infty$ **and** the sequence of set complements $\theset{B_n^c}_{n=1}^\infty$ satisfies
$$
\mu\left(\bigcap_{n=k}^{K} B_{n}^{c}\right)=\prod_{n=k}^{K}\left(1-\mu\left(B_{n}\right)\right)
$$
for all positive integers $k$ and $K$ with $k < K$, then $\mu(B) = 1$.

> Hint: Use the fact that $1 - x \leq e^{-x}$ for all $x$.

## 4. 
Let $\{u_n\}_{n=1}^∞$ be an orthonormal sequence in a Hilbert space $\mathcal{H}$.

a. Prove that for every $x ∈ \mathcal H$ one has 
$$
\displaystyle\sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2} \leq\|x\|^{2}
$$

b. Prove that for any sequence $\{a_n\}_{n=1}^\infty \in \ell^2(\NN)$ there exists an element $x\in\mathcal H$ such that 
  $$
  a_n = \inner{x}{u_n} \text{ for all } n\in \NN
  $$
  and
  $$
  \norm{x}^2 = \sum_{n=1}^{\infty}\left|\left\langle x, u_{n}\right\rangle\right|^{2}
  $$

## 5.

a. Show that if $f$ is continuous with compact support on $\RR$, then 
$$
\lim _{y \rightarrow 0} \int_{\mathbb{R}}|f(x-y)-f(x)| d x=0
$$

b. Let $f\in L^1(\RR)$ and for each $h > 0$ let 
$$
\mathcal{A}_{h} f(x):=\frac{1}{2 h} \int_{|y| \leq h} f(x-y) d y
$$

i. Prove that $\left\|\mathcal{A}_{h} f\right\|_{1} \leq\|f\|_{1}$ for all $h > 0$.

ii. Prove that $\mathcal{A}_h f \to f$ in $L^1(\RR)$ as $h \to 0^+$.