CRN 10459: HW 5

Revision as of 11:42, 28 October 2025

Problem 21. Two Cauchy sequences $(a_n)$ and $(b_n)$ are said to be equivalent if $\lim_{n\to\infty} |a_n-b_n|=0$. We then write $(a_n)\sim (b_n)$.

1. Show that $\sim$ indeed defines an equivalence relation on the set of all Cauchy sequences.
2. Show: If $(a_n)\sim(b_n)$ and $(c_n)\sim(d_n)$, then $(a_n+c_n)\sim (b_n+d_n)$.
3. Show: If $(a_n)\sim(b_n)$ and $(c_n)\sim(d_n)$, then $(a_n\cdot c_n)\sim (b_n\cdot d_n)$.

Problem 22. A Cauchy sequence $(a_n)$ is said to be positive, if for all $k\in\mathbb{N}$ there is an $N\in\mathbb{N}$ such that $a_n>-\frac{1}{k}$ for all $n\geq N$.

1. Show that the sum of two positive Cauchy sequences is positive.
2. Show that the product of two positive Cauchy sequences is positive.

Problem 23. For a set of real numbers $A$, let $A'$ denote the set of its accumulation points. Find a set $A$ such that $((A')')'=\emptyset$, but $(A')'\neq\emptyset$.

Problem 24. Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.

1. Show that $f$ has a limit at $0$.
2. Show that $f$ has a limit at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.)

Problem 25. Let $f:D\to \mathbb{R}$, and $x_0$ be an accumulation point of $D$. Suppose that $f$ has a limit at $x_0$. Show that there is a $\delta>0$ and an $M>0$ such that $|f(x)|\leq M$ for all $x\in D$ satisfying |x-x_0|< $\delta$.