Final Exam

Due Date: Tuesday, May 24 at 11:59 pm

Rules: This is a final exam, not a homework assignment. You must solve the problems entirely on your own, and you should not discuss the problems with any other students in the class. While working on the exam, you should feel free to consult the notes and homework solutions posted on the class web page, as well as any notes you might have related to the class. If you like, you may also consult the following textbooks:

• The Real Numbers and Real Analysis by Ethan Bloch.
• Proofs & Fundamentals: A First Course in Abstract Mathematics by Ethan Bloch.
• Calculus by James Stewart.
• Topology by James Munkres.
• Principles of Mathematical Analysis by Walter Rudin.
• Real & Complex Analysis by Walter Rudin
• A Primer of Lebesgue Integration by H. S. Bear.
• An Introduction to Measure Theory by Terence Tao.

You should not consult any other textbooks or internet sources when working on the exam.

1. Let \((\Omega,\mathcal{E},P)\) be a probability space, let \(X_n\colon \Omega\to[0,\infty)\) be a sequence of random variables, and suppose that \(X_1\geq X_2\geq X_3\geq \cdots\). Given that \(X_n\to 0\) in probability, prove that \(X_n\to 0\) almost surely.
2. Recall that a step function is any function of the form \[ a_1 \chi_{I_1} + \cdots + a_n \chi_{I_n} \] where \(I_1,\ldots,I_n\) are intervals and \(a_1,\ldots,a_n\in\mathbb{R}\). Prove that there exists a Hilbert basis for \(L^2\bigl([0,1]\bigr)\) whose elements are step functions.
3. Let \(\{a_n\}\) be a sequence of positive numbers, and suppose that \[ \sum_{n=1}^\infty a_n^{3} \,=\, 1. \] Find the maximum possible value of the sum \[ \sum_{n=1}^\infty \frac{a_n}{4^n}.\tag*{($*$)} \] You must prove your answer. In particular, you must prove that the sum (\(*\)) is always less than or equal to the maximum value, and you must exhibit a sequence \(\{a_n\}\) for which the sum (\(*\)) attains the maximum value.