CRN 10459: HW 2

Problem 6. 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. 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. Exercise 2.4.7.

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 above. Show that there is a sequence $(x_n)_{n=1}^\infty$ of elements in $X$ that converges to $\sup X$. Note: You have to construct such a sequence, i.e., say how you choose $x_1\in X$, $x_2\in X$, etc.