CRN 10459: HW 5 - Revision history

HelmutKnaust at 16:43, 28 October 2025

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HelmutKnaust at 16:36, 28 October 2025

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HelmutKnaust at 22:07, 27 October 2025

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HelmutKnaust: Created page with "'''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 this indeed defines an equivalence relation ..."

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Created page with "'''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 this indeed defines an equivalence relation ..."

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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 this indeed defines an equivalence relation on the set of all Cauchy sequences.

'''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 $A'\neq\emptyset$. Find a set $B$ such that $((B')')'=\emptyset$, but $(B')'\neq\emptyset$.

'''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|< $\delta$.

← Older revision		Revision as of 16:43, 28 October 2025
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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\|< $\delta$.	+	$\|f(x)\|\leq M$ for all $x\in D$ satisfying $\|x-x_0\|< \delta$.

@@ Line 8: / Line 8: @@
 #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 $A'\neq\emptyset$. Then find a set $B$ such that  $((B')')'=\emptyset$, but $(B')'\neq\emptyset$.
+'''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 $f(x)=\sqrt[3]{x}$.

← Older revision		Revision as of 16:40, 28 October 2025
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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\|< $\delta$.	+	$\|f(x)\|\leq M$ for all $x\in D$ satisfying \|x-x_0\|< $\delta$.

← Older revision		Revision as of 16:36, 28 October 2025
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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 ~~this~~ indeed defines an equivalence relation on the set of all Cauchy sequences.	+	'''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 $n\geq N$.

← Older revision		Revision as of 16:33, 28 October 2025
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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 this indeed defines an equivalence relation on the set of all Cauchy sequences.	+	'''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 this indeed defines an equivalence relation on the set of all Cauchy sequences.

	'''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 $n\geq N$.