Math 105A Homework 4

Please complete the following from the textbook:

4.1.1

Prove the infinum version of theorem 36:

For a set $A$ that is bounded above, a number $\alpha$ equals $\sup A$ if and only if $\alpha$ is an upper bound of $A$, and for all $\varepsilon > 0$, there exists $x \in A$ such that $x > \alpha - \varepsilon$

4.1.2

Prove the infinum version of theorem 37:

Assume that $A$ is a set in the reals that is bounded above, let $\alpha = \sup A$ and assume that $\alpha \notin A$. Then for any $\varepsilon > 0$ there is an infinite number of elements of $A$ contained within the interval $(\alpha - \varepsilon, \alpha)$

4.1.3

Prove that $c < 0$ case of Theorem 38:

For $A$ a bounded set of the reals, we have that

\[\sup(cA) = c \sup A,\quad \inf(cA) = c\inf(A)\]

for a real constant $c > 0$, and

\[\sup(cA) = c\inf(A),\quad \inf(cA) = c \sup A\]

for a real constant $c < 0$

4.2.1

Prove that a bounded monotonically decreasing sequence converges.

4.2.2

Let $A$ be a non-empty subset of $\mathbb R$ that is bounded above, then there exists a sequence ${x_n}$ with $x_n \in A$ for all $n \in \mathbb N$ with ${x_n} \rightarrow \sup A$

4.2.3

Define a sequence ${a_n}$ by $a_1 = \sqrt2$ and $a_{n+1} = \sqrt{2 + a_n}$

  • Show that $a_n \leq 2$ for every $n$
  • Show that ${a_n}$ is an increasing sequence. And then explain why ${a_n}$ converges.
  • Show that $\lim_{n \rightarrow \infty} a_n = 2$

4.2.4

Let $k > 1$ be a constant, and define a sequence ${a_n}$ by $a_1 = 1$ and

\[a_{n+1} = \frac{k(1+a_n)}{k + a_n}\]
  • Show that ${a_n}$ converges. (Either show the sequence is Cauchy or satisfies the conditions in the monotone convergence theorem)
  • Find $\lim_{n \rightarrow \infty}a_n$

4.2.5

From Rudin: fix $\alpha > 1$ and take $x_1 > \sqrt{\alpha}$ and define $x_{n+1} = \frac{\alpha + x_n}{1 + x_n}$

  • Prove that $x_1 > x_3 > x_5 > \dots$
  • Prove that $x_2 < x_4 < x_6 < \dots$
  • Find what ${x_n}$ converges to and prove your claim

4.2.6

For $x_1 > 0$ and $\alpha > 0$ define the sequence ${x_n}$ recursively by

\[x_{n+1} = \frac{p-1}{p}x_n + \frac{\alpha}{px_n^{p-1}}\]

where $p$ is a fixed positive integer. Prove what ${x_n}$ converges to.

Hint: Monotone Convergence theorem and the arithmetic mean vs geometric mean inequality will help here

Additional Question

By quoting 3 things, explain how the Bolzano-Weierstrass Theorem implies the Cauchy Completeness of $\mathbb R$. (i.e. Bolzano-Weierstrass is another equivalent form of completeness when the AP is present)