CRN 11247: HW 2 - Revision history

HelmutKnaust at 17:11, 9 September 2021

2021-09-09T17:11:39Z

HelmutKnaust at 17:10, 9 September 2021

2021-09-09T17:10:38Z

HelmutKnaust at 17:09, 9 September 2021

2021-09-09T17:09:48Z

HelmutKnaust at 17:06, 9 September 2021

2021-09-09T17:06:57Z

HelmutKnaust: Created page with "'''Problem 6.''' #Show that the sets $\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ and $\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ have the same cardinality. #Show t..."

2021-09-09T16:59:55Z

Created page with "'''Problem 6.''' #Show that the sets $\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ and $\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ have the same cardinality. #Show t..."

New page

'''Problem 6.'''
#Show that the sets $\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ and $\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ have the same cardinality.
#Show that $[0,1]$ and $(0,1)$ have the same cardinality. Hint: Problem 6.1 may help.

'''Problem 7.''' Using the limit definition, show that the sequence $(a_n)_{n=1}^\infty$, given by \[a_n=\sqrt{\frac{2n+5}{n+2}}\] converges to $\sqrt{2}$.

'''Problem 8.''' Suppose the sequence $(a_n)_{n=1}^\infty$ converges to $0$, and the sequence $(b_n)_{n=1}^\infty$ is bounded. Show that the sequence $(a_n \cdot b_n)_{n=1}^\infty$ converges to $0$.

'''Problem 9.''' Suppose the sequence $ (a_n)_{n=1}^\infty$ converges to a limit $x$. For $n\in\mathbb{N}$ let \[b_n=\frac{1}{n}\left(a_1+a_2+\cdots +a_n\right).\] Show that the sequence $(b_n)_{n=1}^\infty$ converges to $x$.

'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$.

@@ Line 1: / Line 1: @@
 '''Problem 6.'''
 #Show that the sets $\{\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$ and $\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\}$  have the same cardinality.
-#Show that $[0,1]$ and $(0,1)$ have the same cardinality. Hint: Problem 6.1 may help.
+#Show that $[0,1]$ and $(0,1)$ have the same cardinality. ''Hint:'' Problem 6.1 may help.
 '''Problem 7.''' Using the limit definition, show that the sequence $(a_n)_{n=1}^\infty$, given by \[a_n=\sqrt{\frac{2n+5}{n+2}}\] converges to $\sqrt{2}$.
@@ Line 11: / Line 11: @@
 #Show that the converse is false: Find a sequence $ (a_n)_{n=1}^\infty$ such that $(b_n)_{n=1}^\infty$ converges, while $ (a_n)_{n=1}^\infty$ diverges.
-'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$. Hint: you have to construct such a sequence, i.e., say how you choose $x_1\in X$, $x_2\in X$, etc.
+'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$. ''Hint:'' You have to construct such a sequence, i.e., say how you choose $x_1\in X$, $x_2\in X$, etc.

← Older revision		Revision as of 17:10, 9 September 2021
Line 11:		Line 11:
	#Show that the converse is false: Find a sequence $ (a_n)_{n=1}^\infty$ such that $(b_n)_{n=1}^\infty$ converges, while $ (a_n)_{n=1}^\infty$ diverges.		#Show that the converse is false: Find a sequence $ (a_n)_{n=1}^\infty$ such that $(b_n)_{n=1}^\infty$ converges, while $ (a_n)_{n=1}^\infty$ diverges.

−	'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$. Hint: you have to construct such a sequence, i.e., say how you choose $x_1$, $x_2$, etc.	+	'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$. Hint: you have to construct such a sequence, i.e., say how you choose $x_1\in X$, $x_2\in X$, etc.

← Older revision		Revision as of 17:09, 9 September 2021
Line 11:		Line 11:
	#Show that the converse is false: Find a sequence $ (a_n)_{n=1}^\infty$ such that $(b_n)_{n=1}^\infty$ converges, while $ (a_n)_{n=1}^\infty$ diverges.		#Show that the converse is false: Find a sequence $ (a_n)_{n=1}^\infty$ such that $(b_n)_{n=1}^\infty$ converges, while $ (a_n)_{n=1}^\infty$ diverges.

−	'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$.	+	'''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$. Hint: you have to construct such a sequence, i.e., say how you choose $x_1$, $x_2$, etc.

@@ Line 7: / Line 7: @@
 '''Problem 8.''' Suppose the sequence $(a_n)_{n=1}^\infty$ converges to $0$, and the sequence $(b_n)_{n=1}^\infty$ is bounded. Show that the sequence $(a_n \cdot b_n)_{n=1}^\infty$ converges to $0$.
-'''Problem 9.''' Suppose the sequence $ (a_n)_{n=1}^\infty$ converges to a limit $x$. For $n\in\mathbb{N}$ let \[b_n=\frac{1}{n}\left(a_1+a_2+\cdots +a_n\right).\] Show that the sequence $(b_n)_{n=1}^\infty$ converges to $x$.
+'''Problem 9.'''
 '''Problem 10.''' Let $X$ be a non-empty set that is bounded from below. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\inf X$.