Math 104 Assignment 5

1. Closures and limits

   Let $E$ be a subset of a metric space $(M, d)$. Show that $x \in \overline{E}$ if and only if there is a sequence in $E$ which converges to $x$.

2. Square roots and convergent sequences

   Suppose that $(z_a)_a$ is a sequence in $\R_{\geq0}$ with \[\lim_{s\to\infty} z_s = z.\] Show that $(\sqrt{z_n})_n$ converges to $\sqrt z$. (Hint: it may be useful to treat the case $z = 0$ separately; it may also be useful to use the fact that $(\sqrt{z_t} - \sqrt{z})(\sqrt{z_t}+\sqrt{z}) = z_t - z$.)

3. A recursively defined sequence

   Let $(a_n)_n$ be the sequence defined as follows: $a_1 = 1$, and for $n \in \N$, $a_{n+1} = \sqrt{10+3a_n}$.
   1. Prove that for every $n \in \N$, we have $0 \leq a_n \leq a_{n+1} \leq 10^{100}$.
   2. We will see later that Part A means $(a_n)_n$ must converge (monotonic sequences converge if and only if they are bounded). Find $\lim_{n\to\infty} a_n$.

4. Subsubsequences

   1. Suppose that $(M, d)$ is a metric space and $a \in M$. Suppose further that $(x_k)_k$ is a sequence in $M$ with the property that every subsequence of $(x_k)_k$ has a further subsequence which converges to $a$. Show that $(x_k)_k$ converges to $a$.
   2. Give an example of a sequence $(y_k)_k$ in some metric space, which does not converge but has the property that every subsequence has a further subsequence which does converge.

5. An interesting sequence of rational numbers

   Let $(q_n)_n$ be a sequence in $\Q_{\geq0}$ defined as follows: \[q_n = \begin{cases} \frac{a}{b} & \text{ if } n = 2^a3^b \color{red}{\text{ for some }a, b \in \N} \\ 0 & \text{otherwise} \end{cases}.\] Show that for any $x \in \R_{\geq0}$ there is a subsequence of $(q_n)$ converging to $x$. (Hint: first show that every positive rational number occurs infinitely often in $(q_n)_n$, and recall that $\Q$ is dense in $\R$.)