Sequences and limits

Did you think Olaf would forget that you remain a disgrace to the human species?!
Olaf is here to make your existence marginally less useless!

We have already seen velocity defined as $\frac{x_1 - x_2}{t_1 - t_2}$ as $t_1 - t_2$ becomes smaller and smaller.

We shall now present the necessary mathematics in order to make such a wishy washy wooly and wimpy statement rigorous enough to withstand a barbarian's bachelor's party!

We have already discussed the real numbers $\mathbb{R}$. Today I will introduce the subset $\mathbb{N} \subset \mathbb{R}$ defined by:

$\mathbb{N} = \left\{1,2,3,4,..\right\}$

We shall refer to $\mathbb{N}$ as the set of natural numbers.

Notice that $n \in \mathbb{N} \implies n + 1 \in \mathbb{N}$. This leads to a rich application in mathematics, computer science and mating, as it is the underlying fact behind the techniques of mathematical induction and recursion. However, these will not detain us at this stage.

Consider now an infinite sequence of real numbers, labbeled: $a_1, a_2, a_3,...,a_n,...$. For convenience, the entire sequence may be referred to as $(a_n)_{n \in \mathbb{N}}$

consider the sequence defined by $a_n = \frac{1}{n}, n\in \mathbb{N}$

As $n$ is chosen to be larger and larger, $\frac{1}{n}$ will become smaller.
Indeed, for any real number $\epsilon > 0$, I can find an $N \in \mathbb{N}$ such that $\frac{1}{N} \leq \epsilon$ In this sense, we say that the sequence $(\frac{1}{n})_{n\in \mathbb{N}}$ appraches 0 as $n$ approaches infinity.

We shall generelise this notion, giving the definition of a limit of a sequence:

A sequence $(a_n)_{n\in \mathbb{N}}$ is said to converge to $a$ if for every $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that $|a_n - a| \leq \epsilon$ for all $n \geq N$. $a$ is also known as the limit of the sequence.

We can write this as $\underset{n \rightarrow \infty}{\lim} a_n = a$ or $a_n \underset{n \rightarrow \infty}{\longrightarrow} a$.

if $(b_n)_{n \in \mathbb{N}}$ is some other sequence which converges to $b$, then then we have the following result, whose proof would be up to you to produce! FOR ALL BARBARIANS PRODUCE THEIR OWN PROOFS!

$\underset{n \rightarrow \infty} {\lim} (a_n + b_n) = \underset{n \rightarrow \infty} {\lim} a_n + \underset{n \rightarrow \infty} {\lim} b_n = a+b$

$\underset{n \rightarrow \infty} {\lim} (a_n \times b_n) = \underset{n \rightarrow \infty} {\lim} a_n \times \underset{n \rightarrow \infty} {\lim} b_n = ab$

$\underset{n \rightarrow \infty} {\lim} (\frac{a_n}{b_n}) = \frac{\underset{n \rightarrow \infty} {\lim} a_n} {\underset{n \rightarrow \infty} {\lim} b_n} = \frac{a}{b}$ if $b \neq 0$

We are now equipped with the proper weapons to attack calculus!

But you now require rest. GIVE ME 1000 PUSHUPS NOW, WEAKLINGS