Problem Set Generator (LaTeX)

\documentclass[11pt]{article} \usepackage[margin=1in]{geometry} \usepackage{amsmath, amssymb, amsthm} \usepackage{enumitem} \usepackage{xcolor} \usepackage{hyperref} \newcounter{problemnum} \newenvironment{problem}[1]{% \refstepcounter{problemnum}% \noindent\textbf{Problem #1.}\quad }{\vspace{1em}} \newenvironment{solution}{% \noindent\textit{\color{teal!70!black}Solution.}\quad }{\vspace{1em}} \title{Introduction to Real Analysis \ -- \ Problem Set 3} \author{MATH 131A} \date{Due: October 15, 2026} \begin{document} \maketitle \noindent\textbf{Topic:} Sequences, limits, and continuity \bigskip \begin{problem}{1} Prove that the sequence $a_n = (1 + 1/n)^n$ is monotonically increasing and bounded above by 3. Conclude that $a_n$ converges. \end{problem} \begin{solution} We show $a_n < a_{n+1}$ by applying the binomial theorem to both sides and comparing term by term. The bound follows from the geometric series estimate $\sum_{k=0}^n \frac{1}{k!} \le 1 + \sum_{k=1}^n \frac{1}{2^{k-1}} < 3$. \end{solution} \begin{problem}{2} Let $f: [a, b] \to \mathbb{R}$ be continuous. Prove that $f$ attains its maximum on $[a, b]$. \end{problem} \begin{solution} By boundedness of continuous functions on compact intervals, $M = \sup f$ exists and is finite. Take a sequence $x_n$ with $f(x_n) \to M$; by Bolzano-Weierstrass, $x_n$ has a convergent subsequence $x_{n_k} \to x^* \in [a, b]$. By continuity, $f(x^*) = M$. \end{solution} \begin{problem}{3} Suppose $f_n \to f$ pointwise on $[0, 1]$ and each $f_n$ is continuous. Give an example where $f$ is not continuous. \end{problem} \begin{solution} Take $f_n(x) = x^n$. Then $f_n \to f$ pointwise where $f(x) = 0$ for $x < 1$ and $f(1) = 1$, which is discontinuous at $x = 1$. \end{solution} \end{document}