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## Math 330 - 02 Homework

• LaTeX-ed solutions are encouraged and appreciated.
• If you use LaTeX, hand-in a printed version of your homework.
• You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange of any written material.
• Writing of homework problems should be done on an individual basis.
• References to results from the textbook and/or class notes should be included.
• The following lists should be considered partial and tentative lists until the word complete appears next to it.
• Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.


Problem Set 10 (complete) Due: 11/17/2017. Board Presentation: 11/17/2017

1. Prove Prop. 10.23.v
2. Prove The. 10.26
3. Let $(a_n)$ be a sequence. Consider the sequence of even-indexed terms, $(a_{2n})$, and the sequence of odd-indexed terms, $(a_{2n+1})$. Prove that if both $(a_{2n})$ and $(a_{2n+1})$ converge to $L$, then $(a_n)$ converges to $L$.
4. Let $q_n=\displaystyle\frac{f_n}{f_{n+1}}$, where $f_n$ is the $n$-th Fibonacci number. Show that the sequence $(q_n)$ converges. What value does it converge to?

Problem Set 9 (complete) Due: 11/10/2017. Board presentation: 11/10/2017

1. Prove Prop. 10.10.iii
2. Prove Prop. 10.17
3. Prove Prop. 10.23.iii

Problem Set 8 (complete) Due: 11/03/2017. Board presentation: 11/03/2017

1. Prove Prop. 8.50
2. Prove that function composition is associative, when defined.
3. Let $A,B,C$ be sets and $f:A\to B$ and $g:B\to C$ functions. Prove that if $g\circ f$ is surjective then $g$ is surjective. Give an example when $g\circ f$ is surjective, but $f$ is not.
4. Construct an example of a function with several right inverses.
5. Prove Prop. 9.15 (Hint: induction on $k$)
6. Prove Prop. 9.18