$\newcommand{\br}{\\}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\N}{\mathbb{N}}$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$
$\newcommand{\L}{\mathcal{L}}$
$\newcommand{\spa}[1]{\text{span}(#1)}$
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\emptyset}{\varnothing}$
$\newcommand{\otherwise}{\text{ otherwise }}$
$\newcommand{\if}{\text{ if }}$
$\newcommand{\proj}{\text{proj}}$
$\newcommand{\union}{\cup}$
$\newcommand{\intercept}{\cap}$
$\newcommand{\abs}[1]{| #1 |}$
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$
$\newcommand{\pare}[1]{\left\(#1\right\)}$
$\newcommand{\t}[1]{\text{ #1 }}$
$\newcommand{\head}{\text H}$
$\newcommand{\tail}{\text T}$
$\newcommand{\d}{\text d}$
$\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$
$\newcommand{\inv}[1]{{#1}^{-1}}$
$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$
$\newcommand{\nullity}[1]{\text{nullity}(#1)}$
$\newcommand{\rank}[1]{\text{rank }#1}$
$\newcommand{\ceil}[1]{\lceil#1\rceil}$
$\newcommand{\floor}[1]{\lfloor#1\rfloor}$
$\newcommand{\var}[1]{\text{var}(#1)}$
$\newcommand{\tr}[1]{\text{tr}(#1)}$
$\newcommand{\oto}{\text{ one-to-one }}$
$\newcommand{\ot}{\text{ onto }}$
$\newcommand{\Re}[1]{\text{Re}(#1)}$
$\newcommand{\Im}[1]{\text{Im}(#1)}$
$\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$
$\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$
$\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$
$\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$
$\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$
$\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$
$\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$
$\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$
$\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$
$\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Definition: Sequence
A sequence is function whose domain is a set of the form $\set{ n \in \Z \mid n \geq m } $; $m =0$ or $m = 1$. We denote the sequence by a letter such as $s $ and denote its value at $n $ as $s_n $.
Definition: Limit
A sequence $(s_n) $ of real numbers is said to converge to the real number $s $ provided that
for each $\epsilon > 0$ there exists a number $N $ such that
$$n > N \to \abs{ s_n -s } < \epsilon $$
If $(s_n) $ converges to $s$, we will write $\limu{ n }{ \infty }s_n = s $, or $s_n \to s $.
The number $s $ is called the limit of the sequence $(s_n) $.
A sequence that does not converge to some real number is said to diverge.
Example
$s_n = \frac{ 1 }{ n }, \limu{ n }{ \infty } s_n = 0 $
Proof