CRN 11378: HW 5

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Problem 21. Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.

  1. Show that $f$ is continuous at $0$.
  2. Show that $f$ is continuous at any $x_neq0$. (The identity $a^3-b^3=)a-b)(a^2+ab+b^2)$ will be helpful.)

Problem 17. Prove: A subset $F\subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ converges to an element in $F$.

Problem 18. Find all accumulation points of the set \[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\] Remember that $A=B\ \Leftrightarrow\ (A\subseteq B)\wedge (B\subseteq A)$.

Problem 19. Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.

Problem 20. Consider the following sets: \[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\] For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.

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