CRN 11247: HW 2
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'''Problem 6.''' | '''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 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 $\sqrt{2}$. | '''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 $\sqrt{2}$. | ||
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'''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 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.''' |
| + | #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 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 $x_1\in X$, $x_2\in X$, etc. |
Latest revision as of 11:11, 9 September 2021
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 $x$.
- 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.
