CRN 10459: HW 2 - Revision history

HelmutKnaust at 16:18, 15 September 2025

2025-09-15T16:18:15Z

HelmutKnaust at 16:12, 15 September 2025

2025-09-15T16:12:12Z

HelmutKnaust: Created page with "'''Problem 6.''' 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 7.''' ..."

2025-09-15T16:11:39Z

Created page with "'''Problem 6.''' 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 7.''' ..."

New page

'''Problem 6.''' 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 7.''' 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 8.''' Exercise 2.4.7.

'''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 $x$.
#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 above. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\sup X$. ''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 16:18, 15 September 2025
Line 1:		Line 1:
−	'''Problem 6.''' 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 6.''' 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 7.''' 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 7.''' 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 $0$.

← Older revision		Revision as of 16:12, 15 September 2025
Line 9:		Line 9:
	#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 above. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\sup X$. ''~~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 above. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\sup X$. ''Note:'' You have to construct such a sequence, i.e., say how you choose $x_1\in X$, $x_2\in X$, etc.