CRN 11247: 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. 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.

1. 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 $x$.
2. 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 $\inf X$.