CRN 11982: HW 2
From Classes
(Difference between revisions)
HelmutKnaust (Talk | contribs) (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...") |
HelmutKnaust (Talk | contribs) |
||
| Line 3: | Line 3: | ||
#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.''' | + | '''Problem 7.''' Suppose the sequence $(a_n)_{n=1}^\infty$ converges to both $L$ and $M$. Show that $L=M$. (Limits are unique.) |
'''Problem 8.''' Exercise 2.3.7(a)(b) | '''Problem 8.''' Exercise 2.3.7(a)(b) | ||
Latest revision as of 11:13, 8 September 2014
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. Suppose the sequence $(a_n)_{n=1}^\infty$ converges to both $L$ and $M$. Show that $L=M$. (Limits are unique.)
Problem 8. Exercise 2.3.7(a)(b)
Problem 9. 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 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 $\inf X$.
