% This is lecture no. 4 of MH3100/: Real Analysis I.

\chapter{Sequences \& Limits}%
\label{cha:sequences_&_limits}

\section{Sequences \& Limits}%
\label{sec:sequences_&_limits}

\begin{exmp}
	Newton's Method.
	Newton's Method is used to find the zeros of a function $f\vcentcolon \mathbb{R}\mapsto\mathbb{R}$ by generating a sequence $\left\{x_n\right\}$ via the recurrence
	\[
	x_{n+1}\defas x_n - \frac{f(x_n)}{f^{\prime}(x_n)}
	.\]
	For example, applying Newton's method to the function $f\vcentcolon x\in\mathbb{R}\mapsto x^2-a$ for some number $a\in\mathbb{R}$ gives the Babylonian Method for finding $\sqrt{a}$:
	\[
	x_{n+1}\defas \frac{1}{2}\left(x_n + \frac{a}{x_n}\right)
	.\]
\end{exmp}

\begin{exmp}
	Taylor Series.
	The Taylor series of a function $f\vcentcolon D\subseteq\mathbb{R}\mapsto\mathbb{R}$ centered at $a\in D$ is
	\[
	\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^{n}
	.\]
	For each $x\in\mathbb{R}$, the above infinite sum is a real number if the sequence of partial sums $\left\{s_n\right\}$ given by
	\[
	s_n \defas \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^{k}
	\]
	is a convergent sequence.
\end{exmp}

\begin{definition}
	Sequences.
	A \textbf{sequence} of real numbers is a function from $\mathbb{N}$ to $\mathbb{R}$.
	We usually write a sequence as
	\[
	a_1,a_2,a_3,\dotsc,a_n,\dotsc
	\]
	where $a_n$ denotes the image of the number $n\in\mathbb{N}$.
	For each $n\in\mathbb{N}$, the number $a_n$ is called a \textbf{term} of the sequence.
	We use the notation $\left\{a_n\right\}^{\infty}_{n=1}$ (or simply $\left\{a_n\right\}$) to denote the above sequence.
\end{definition}

\bigbreak
\noindent
In this module, the following terms are used when dealing with sequences.
\begin{enumerate}
	\item A \textbf{tail} of a sequence $\left\{a_n\right\}$ is the sequence
		\[
		a_N, a_{N+1}, a_{N+2},\dotsc
		\]
		for some number $N\in\mathbb{N}$.
	\item A sequence is said to satisfy a property \textbf{eventually} if the property is satisfied by every term in  a tail of the sequence.
\end{enumerate}

\begin{definition}
	Limits of Sequences.
	The sequence $\left\{a_n\right\}$ is said to have the \textbf{limit} $a\in\mathbb{R}$ if
	\[
	\forall\varepsilon>0,\ \exists N\in\mathbb{N}\backepsilon\forall n\in\mathbb{N}\quad
	n > N\implies |a_n-a|<\varepsilon
	.\]
	We use the notation $\lim_{n \to \infty}a_n$ (or simply $\lim a_n$) to denote the limit, and say that $\left\{a_n\right\}$ \textbf{converges} to $a$.
	We write $a=\lim_{n \to \infty}a_n$ to mean that $\left\{a_n\right\}$ has the limit $a$.
	We say that a sequence $\left\{a_n\right\}$ is \textbf{convergent} if it has a limit.
	Otherwise, we say that $\left\{a_n\right\}$ is \textbf{divergent} and that it \textbf{diverges}.
\end{definition}

\section{Convergent Sequences \& Their Limits}%
\label{sec:convergent_sequences_&_their_limits}

\begin{thrm}
	Uniqueness of Limits.
	A \textbf{convergent} sequence $\left\{a_n\right\}$ has a \textbf{unique limit}.
\end{thrm}

\begin{proof}
	Let $a$ and $a^{\prime}$ be limits of $\left\{a_n\right\}$.
	The definitions of $a=\lim_{n \to \infty}a_n$ and $a^{\prime} = \lim_{n \to \infty}a_n$ imply that $\forall\varepsilon>0$ 
	\[
	\left\{a_n\right\}\text{ eventually satisfies }|a_n - a|<\varepsilon\text{ and }|a_n-a^{\prime}|<\varepsilon
	.\]
	Therefore $\forall\varepsilon>0$ 
	\begin{align*}
		\left\{a_n\right\}\text{ eventually satisfies }|a - a^{\prime}| &\leqslant |a-a_n| + |a_n - a^{\prime}|\\
										& < \varepsilon +\varepsilon\\
										& = 2\varepsilon.
	\end{align*}
	Hence $|a-a^{\prime}|<2\varepsilon$ for all $\varepsilon>0$, which shows that $a=a^{\prime}$.
\end{proof}

\begin{definition}
	Bounded Sequence.
	A sequence $\left\{a_n\right\}$ is said to be \textbf{bounded} if there exists $M>0$ such that
	\[
	\forall n\in\mathbb{N}\quad |a_n|\leqslant M
	.\]
	Otherwise the sequence is said to be \textbf{unbounded}.
\end{definition}

\begin{thrm}
	Eventual Boundedness.
	A sequence $\left\{a_n\right\}$ is \textbf{bounded} iff it is \textbf{eventually bounded}, that is, its tail has all its terms satisfying $|a_n|\leqslant M$ for some $M>0$.
\end{thrm}

\begin{proof}
	There is some number $M>0$ and a tail $\left\{a_n\right\}^{\infty}_{n=N}$ where $\forall n\in\mathbb{N}$ 
	\[
	|a_{N+n}|\leqslant M
	.\]
	Therefore, $\forall n\in\mathbb{N}$ 
	\[
	|a_n|\leqslant\max\left\{|a_1|,|a_2|,\dotsc,|a_N|,M\right\}
	\]
	shows that $\left\{a_n\right\}$ is bounded.
\end{proof}

\begin{thrm}
	Boundedness of Convergent Sequences.
	Every \textbf{convergent} sequence $\left\{a_n\right\}$ is \textbf{bounded}.
\end{thrm}

\begin{proof}
	Let $a$ denote the limit of $\left\{a_n\right\}$.
	The definition of $a=\lim_{n \to \infty}a_n$ implies that for $\varepsilon=1>0$ 
	\begin{align*}
		\exists N\in\mathbb{N}\backepsilon\forall n>N\quad |a_n-a|<1 & \implies a-1<a_n<a+1\\
									     & \implies -|a|-1<a_n<|a|+1\\
									     & \implies |a_n|<|a|+1.
	\end{align*}
	This shows that $\left\{a_n\right\}$ is eventually bounded, when convergent.
\end{proof}
