By far the main piece of work for the assignment is the Main Report.

• Use the various forecasting methods developed in the course to make forecasts on your dataset
• Evaluate those models as forecasting models for your dataset
• Identify which model makes the ‘best’ point-forecasts
• Make some density/interval and 1-year ahead forecasts
• Due the day before your new data is released; new data must be released between the 1st and 12th of November.
• Usually comes to about 9 or 10 pages $+$ code

## True or False

Just circle true or false. Each equestion 3 points.

• TF-every finite ordered set has a minimum
• TF – The sequence $1,0,2,0,4,0,8,0, \ldots$ converges to $\infty$ in the extended reals
• T F – The rational numbers satisfy the least upper bound property
• TF – The set of odd integers is countable
• TF If $a_1, a_2, a_3, \ldots$ is a Cauchy sequence of positive numbers with $\alpha_i>0$ for all $i$, then $\lim _{i \rightarrow \infty} a_i>0$.

## Multiple Choice

1. Which of these subsets is open in $\mathbb{R}$ ? (with the usual metric $d(x, y)=|x-y|$.) Mark all that apply.
a. $\mathbb{R}$
b. $[0,1)$
c. $\mathbb{Q}$
d. The infinite union $(0,1) \cup(2,3) \cup(4,5) \cup(6,7) \ldots$
2. Let $X$ be the metric space of all bounded functions on $\mathbb{R}$, with the sup metric $d(f, g)=|f-g|_{\text {sup }}$, and let $E$ be the open ball $B(0,1)$ of radius 1 centered at the function 0 . Which of the following functions are accumulation points of $E$ in $X$ ? Mark all that apply.
a. The constant function $f(x)=0$
b. The constant function $f(x)=1$
c. $f(x)=x$
d. $\mathrm{f}(\mathrm{x})=(2 / \pi) \arctan x$
3. Which of the following sequences of real numbers is a Cauchy sequence? Mark all that apply.

a. $1,-1,1,-1,1,-1, \ldots$
b. $1,1 / 2,1 / 4,1 / 8,1 / 16, \ldots$
c. $1,-1 / 2,1 / 4,-1 / 8,1 / 16 \ldots$
d. $1,1,1,1,1,1,1, \ldots$

1. Explain to someone who is not taking real analysis what an equivalence relation is and why we are interested in them
2. Let $v$ be the vector $(1,-2)$ in $\mathbb{R}^2$. Compute $|v|_1,|v|_2$ and $|v|_{\infty}$.
3. Write down a bijection from the integers to the odd integers. (no need to prove it is a bijection.)
4. Explain why a “norm” on $\mathbb{R}^2$ that gives a open ball that looks like $a *$ is not a norm
5. Let $E$ be a nonempty subset of the reals which is bounded. Prove that sup $E$ does not lie in the interior of $E$

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# MATH 350Real AnalysisFall 2022 (also offered Spring 2023)Division III Quantative/Formal Reasoning

CATALOG SEARCH

## Class Details

Why is the product of two negative numbers positive? Why do we depict the real numbers as a line? Why is this line continuous, and what do we mean when we say that? Perhaps most fundamentally, what is a real number? Real analysis addresses such questions, delving into the structure of real numbers and functions of them. Along the way we’ll discuss sequences and limits, series, completeness, compactness, derivatives and integrals, and metric spaces. Results covered will include the Cantor-Schroeder-Bernstein theorem, the monotone convergence theorem, the Bolzano-Weierstrass theorem, the Cauchy criterion, Dirichlet’s and Riemann’s rearrangement theorem, the Heine-Borel theorem, the intermediate value theorem, and many others. This course is excellent preparation for graduate studies in mathematics, statistics, and economics.

The Class:Format: lecture
Limit: 40
Expected: 25
Class#: 1467
Grading: no pass/fail option, yes fifth course option

Requirements/Evaluation:Problem sets, exams, and an expository essay.

Prerequisites:MATH 250 or permission of instructor.

Enrollment Preferences:Juniors and Seniors.

Distributions:Division IIIQuantative/Formal Reasoning

QFR Notes:It’s math.

Updated 10:33 AM