CRN 10459: HW 5
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| − | '''Problem 21.''' Two Cauchy sequences $(a_n)$ and $(b_n)$ are said to be equivalent if $\lim_{n\to\infty} |a_n-b_n|=0$. Show that | + | '''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 $(a_n)\sim (b_n)$. |
| + | #Show that $\sim$ indeed defines an equivalence relation on the set of all Cauchy sequences. | ||
| + | #Show: If $(a_n)\sim(b_n)$ and $(c_n)\sim(d_n)$, then $(a_n+c_n)\sim (b_n+d_n)$. | ||
| + | #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 $n\geq N$. | '''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 $n\geq N$. | ||
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#Show that the product of two positive Cauchy sequences is positive. | #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 | + | '''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 $f(x)=\sqrt[3]{x}$. | '''Problem 24.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$. | ||
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'''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 | '''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|< | + | $|f(x)|\leq M$ for all $x\in D$ satisfying $|x-x_0|< \delta$. |
Latest revision as of 11:43, 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 $(a_n)\sim (b_n)$.
- Show that $\sim$ indeed defines an equivalence relation on the set of all Cauchy sequences.
- Show: If $(a_n)\sim(b_n)$ and $(c_n)\sim(d_n)$, then $(a_n+c_n)\sim (b_n+d_n)$.
- 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 $n\geq N$.
- Show that the sum of two positive Cauchy sequences is positive.
- 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 $f(x)=\sqrt[3]{x}$.
- Show that $f$ has a limit at $0$.
- 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 $|x-x_0|< \delta$.
