CRN 12109: HW 2
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Latest revision as of 10:16, 10 September 2013
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. Exercise 2.2.7(a)
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 $\inf X$.
