# Sequences by Peter

**Definition 1.** $\{x_n\}$ is constant if $x_n=x_{n+1}$ $\forall n\in {\mathbb N}$.

**Example 1.** $\{17\}$ is constant since $x_n=17=x_{n+1}$ $\forall n\in {\mathbb N}$.

**Counterdefinition 1.** $\{x_n\}$ is not constant if $\exists n\in {\mathbb N}$ such that $x_n \neq x_{n+1}$.

**Example 1'.** $\{(-1)^n\}$ is not constant since ${x_1} = -1 \neq 1 = {x_2}$.

**Definition 2.** $\{x_n\}$ is bounded above if $\exists K\in{\mathbb R}$ such that $x_n\leq K$
$\forall n\in {\mathbb N}$.

**Example 2.** Prove that $\{(-1)^n\}$ is bounded above.
Proof: Let $K=4$. $max\{{(-1)^n}\}=1<4=K.$

**Counterdefinition 2.** $\{{x_n}\}$ is not bounded above if $\forall K\in {\mathbb R}, \exists n\in {\mathbb N}$ such that $x_n\geq K$.

**Example 2'.** $\{{2n}\}$ is not bounded.
**Proof:** Given any $K\in {\mathbb R},$ let $m=\lceil \lvert K \rvert \rceil$.
Then $x_m = 2m = 2\lceil \lvert K \rvert \rceil \geq K$.

**Definition 3.** $\{{x_n}\}$ is bounded below if $\exists K\in {\mathbb R}$ such that $x_n \geq K \forall n\in {\mathbb N}$.

**Example 3.** ${\{(-1)^n}\}$ is bounded below.
**Proof:**
Let $K=-7$. $min{\{(-1)^n}\} =-1 \geq -7$.

**Counterdefinition 3.** ${\{x_n}\}$ is not bounded below if $\forall K\in {\mathbb R}, \exists n\in {\mathbb N}$ such that ${\{x_n}\} \geq K$.

**Example 3'.** ${\{3n}\}$ is not bounded below.
**Proof:** Given any $K\in {\mathbb R}$, let $m=\lceil \lvert K \rvert \rceil$.
Then ${\{x_m}\} = -3m = -3\lceil \lvert K \rvert \rceil \leq K.$

**Definition 4.**

Continue similarly with bounded below and bounded: definition, example, counterdefinition, example. Show all your work.

**Week 2**

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1. Prove that if $\{x_n\}$ is infinitely large then $\{x_n\}$ is unbounded
2. Prove that $\{(-1)^nn\}$ is inf. large
3. Counter the definition of infinitely large.
4. Prove that $$x_n=\begin{cases} n,&\text{if $n$ is even}\\ 0,& \text{if $n$ is odd}\end{cases}$$is not infinitely large, but unbounded
5. Counter the definition of $\lim x_n=L$.
6. Prove that $\lim(-1)^n\neq 0$
7. Prove that $\lim(-1)^n\neq 1$

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**Week 3**

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1. Negate and justify how we negate the following statement

$\forall \epsilon>0, ~~~~\epsilon$ is a bad number.

In particular, why we do not change the inequality $\epsilon>0$.

2. Let $A\subset \mathbb R$. State the definition of $m=\inf~A.$

3. Prove that $\sup\{1-\frac{1}{n}\}=1$

4. Prove that $\inf\{1-\frac{1}{n}\}=0$

5. Prove that $\inf\{(0,1)\}=0$

6. Let $A=\{0\}\cup \{(1,2)\}$. Prove that $\inf A=0$.

7. Ex. 2.1.3 and 2.1.4 in our text. You must prove your answers, not just provide them.

8. Redo Week 1 HW if you want to (for half of the remaning points)

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**Week 4**

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1. Prove that any subsequence of an infinitely large sequence is infinitely large.

2. p. 50 Ex. 2.1.7

3. p. 50 Ex. 2.1.10

4. p. 50 Ex. 2.1.13

5. p. 50 Ex. 2.1.16

6. p. 50 Ex. 2.1.17
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**Week 7**

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1. Without using Lemma on nested Intervals prove that
$$\bigcap_{n=1}^\infty[-1/n, 1/n]=\{0\}.$$
2. Exercise 2.3.5
3. Prove that every unbounded sequence, contains an infinitely large subsequesnce. The proof goes by construction.