# Fall 2010 Final Student-provided questions and solutions to the Fall 2010 final exam.

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## 1Question 1¶

(i) The Archimedean property states that the set of natural numbers $\mathbb N$ is unbounded in $\mathbb R$. Assuming this property, show that $\inf \{1/n: n \in \mathbb N\} = 0$. Conclude that if $t > 0$, there exists some $n \in \mathbb N$ with $\frac{1}{n} < t$, and if $y > 0$, there exists a natural number $n$ such that $n-1 \leq y < n$.
(ii) Use (i) to show that for any two real numbers $x$ and $y$ with $x < y$, there exists a rational number $r$ with $x < y < y$.

Coming soon

## 2Question 2¶

(i) Let $f : A \to \mathbb R$ be a function. Define what it means for $f$ to be continuous on $A$, and uniformly continuous on $A$. Comment on the difference between these definitions.
(ii) Show that if $f$ and $g$ are uniformly continuous functions on $A$, and they are both bounded on $A$, then their product $fg$ is uniformly continuous on $A$.
(iii) Show that if $A$ is the closed bounded interval $A = [a, b]$, then a continuous function $f: A \to \mathbb R$ must be bounded. State any theorems you use. Conclude that the product of two uniformly continuous functions defined on a closed bounded interval is uniformly continuous.
(iv) Give an example of a set $A$ and a uniformly continuous function $f: A \to \mathbb R$ which is not bounded, justifying your choice.

Coming soon

## 3Question 3¶

(i) Let $a \subset \mathbb R$ be a nonempty set. Define what it means for a real number to be the supremum of $A$. State the completeness axiom for $\mathbb R$.

Now let $I_n = [a_n, b_n]$, $n\in \mathbb N$ be a nested sequence of intervals, i.e., $I_{n+1} \subset I_n$ for each $n \in \mathbb N$.

(ii) Show that the set $\{a_n:n \in \mathbb N\}$ is bounded above, and let $a*$ be its supremum.
(iii) Show that $\displaystyle a* \in \cap_{n=1}^{\infty} I_n$.
(iv) If $f: [a, b] \to \mathbb R$ is a continuous function, with $f(a) < 0$ and $f(b) > 0$, show that there exists $x \in (a, b)$ with $f(x) = 0$. Hint: use part (iii).
(v) Show that the polynomial $f(x) = x^4 + 7x^3 - 9$ has at least 2 real roots. Describe how you would locate one of these roots with error less than 0.005.

Coming soon

## 4Question 4¶

Prove or disprove each of the following statements.

(i) If every subsequence of $(x_n)$ has a subsequence that converges to 0, then $\displaystyle \lim_{n \to \infty} x_n = 0$.
(ii) The sequence $(1 + (-1)^n)$ is a Cauchy sequence.
(iii) The sequence $((3n)^{\frac{1}{2n}})$ diverges to $\infty$.

### 4.1Solution¶

(i) I have no idea. That's interesting.
(ii) Disprove.
(iii) Disprove. It converges to 1 for some reason. Why???

## 5Question 5¶

(i) If $0 < x < 1$, show that $x^n \to 0$ as $n \to \infty$.
(ii) If $0 < a <b$, find the limit of the sequence $\displaystyle \left ( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \right )$, stating any theorems that you use.

### 5.1Solution¶

(ii) This is straight out of the textbook. Coming soon.

## 6Question 6¶

(i) Suppose $f: [a, b] \to \mathbb R$ with $c \in (a, b)$. Show that $f$ is differentiable at $c$ if and only if there exists a function $\phi: [a, b] \to \mathbb R$ such that $\mathbb \phi$ is continuous at $c$ and satisfies

$$f(x) - f(c) = \phi(x)(x-c)$$

whenever $x \in [a, b]$. Show that in this case, $f'(c) = \phi(c)$.
(ii) If $f(x) = x^3$, find $\phi(x)$ and deduce that $f'(c) = 3c^2$.
(iii) Suppose $f$ is strictly monotone increasing and continuous on $[a, b]$. Let $J = f([a, b])$ and consider the function $g: J \to [a, b]$ defined as $g = f^{-1}$. Explain why $J$ is a closed bounded interval. Show that if $f$ is differentiable at $c \in [a, b]$ and $f'(c) \neq 0$, then $f^{-1}$ is differentiable at $d = f(c)$ and $g'(d) = \frac{1}{f'(c)}$. Hint: use part (i). Then, using part (ii), if $f(x) = x^3$, find $g'(d)$.

Coming soon

## 7Question 7¶

Let $f$, $g$ be differentiable on $\mathbb R$ and suppose that $f(0) = g(0)$ and $f'(x) \leq g'(x)$ for all $x \geq 0$. Show that $f(x) \leq g(x)$ for all $x \geq 0$. State completely any theorems that you use.

Coming soon