Due Thursday Sep 9th, 6pm.

In the following, a sequence $(a_n)$ means $(a_n)_{n=0}^\infty$, unless otherwise specified. You can only use properties of real number proved in Tao's book, section 5.4.

1. Let $(a_n)$ be a sequence in $\Q$. Suppose there is a rational $0 < r < 1$, such that $|a_{n+1} - a_n| < r |a_n - a_{n-1}|$, prove that $(a_n)$ is a Cauchy sequence.

2. Let $a_0$ be any positive integer, and $a_{n+1} = 3 + \frac{1}{a_n}$. Prove that the $(a_n)$ is a Cauchy sequence. (Hint: use the previous problem)

3. Let $(a_n)$ be a Cauchy sequence in $\Q$, and let the sequence $(b_n)$ be defined such that $b_n = a_{2n}$. Prove that $(b_n)$ is equivalent to $(a_n)$.

4. If $(a_n)$ and $(b_n)$ are Cauchy sequences, prove that $(a_n b_n)$ is also a Cauchy sequence.

5. If $(a_n)$, $(c_n)$ and $(b_n)$ are Cauchy sequences, and $(a_n) \sim (c_n)$, prove that $(a_n b_n) \sim (c_n b_n)$.

(Problem 4 and 5 together proves Tao proposition 5.3.10, multiplication of real are well-defined).