# Sequences by Carolyn
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**Week 1**
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**Definition 1.** $\{x_n\}$ is constant if $x_n=x_{n+1}~\forall ~n~\in~{\mathbb N}$.

**Example 1.** $\{17\}$ is constant since

**Counterdefinition 1.** $\{x_n\}$ is not constant if

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

**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.

**Counterdefinition 2.** $\{x_n\}$ is not bounded above if

**Example 2'.** Prove that $\{2^n\}$ is not bounded above.

**Definition 3.** $\{x_n\}$ is bounded below if

**Example 3.** Prove that $\{(-1)^n\}$ is bounded below.

**Counterdefinition 3.** $\{x_n\}$ is not bounded below if

**Example 3'.** Prove that $\{2n\}$ is not bounded below.

**Definition 4.** $\{x_n\}$ is bounded if $\exists$ K $\in~{\mathbb R}$ such that $\mid x_n\mid$ $\leq$ K $\forall~n~\in~{\mathbb N}$.

**Example 4.** Prove that {$\sin n$} is bounded.

**Counterdefinition 4.** $\{x_n\}$ is not bounded if

**Example 4'.** Prove that $\{4n\}$ is not bounded.

**Additional Homework**
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
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**Week 2**
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**5. Counter the definition of $\lim x_n=L$.**

**6. Prove that $\lim(-1)^n\neq 0$.**

**7. Prove that $\lim\frac{1}{n^2}$=0.**