CRN 11982: HW 2

Problem 6.

1. 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.
2. 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 $\inf X$.