HelmutKnaust at 02:22, 10 September 2019

2019-09-10T02:22:16Z

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..."

2019-09-10T02:20:48Z

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..."

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'''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 $(a_n)$ converges to $0$, and the sequence $(b_n)$ is bounded. Show that the sequence $(a_n \cdot b_n)$ converges to $0$.

'''Problem 9.''' Suppose the sequence $ (a_n)$ 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 $(b_n)$ 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 5: / Line 5: @@
 '''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 $(a_n)$ converges to $0$, and the sequence $(b_n)$ is bounded. Show that the sequence $(a_n \cdot b_n)$ converges to $0$.
+'''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)$ 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 $(b_n)$ converges to $x$.
+'''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 $\inf X$.

CRN 11378: HW 2 - Revision history

HelmutKnaust at 02:22, 10 September 2019

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..."