\chapter{Sequences of functions}
- sequence of functions

\section{Convergence of sequences of functions}
\subsection{Pointwise convergence}
- pointwise convergent sequence of functions
- pointwise Cauchy sequence of functions
if f and g converge uniformly, so does af+bg
\subsection{Uniform convergence}
- uniformly convergent sequence of functions
- uniformly Cauchy sequence of functions


Uniform convergence is a very nice property; we can get approximations of the limiting function with arbitrarily small maximum error given that enough terms are taken. The rest of this part will

- uniform convergence implies pointwise convergence
- Dini's theorem

- if f and g converge uniformly, so does af+bg
- if f and g converge uniformly on compact space, so does fg
- sequence of continuous functions with uniform convergence has continuous limit function

\subsection{Weierstrass M test}

\begin{theorem}[Weierstrass M test]
Consider a sequence of functions $(f_n)_{n \in \mathbb{N}}$ and a positive real sequence $(M_n)_{n \in \mathbb{N}}$.  If for any $x \in U$ we have $|f_n(x)| \leq M_n$ and $\sum^{\infty}_{n=1} M_n$ converges, then $\sum^{\infty}_{n=1}f_n$ converges absolutely and uniformly on $U$.
\end{theorem}

\section{Swapping limits and operators}


Numerous examples of sequences of functions where the limit cannot be passed through the integral. One naturally wonders which conditions allow integrals and limits to be swapped?





- uniform convergence can swap limit with integral
\[\lim_{n\to \infty } \int^{b}_{a} f_n (x) dx =   \int^{b}_{a} \lim_{n \to \infty} f_n (x) dx \]


\begin{theorem}[Bounded convergence theorem (Riemann integral)]
Let$f_n$ be a sequence of real functions uniformly bounded and $f_n \to f$ pointwise if all the $f_n$ and $f$ are Riemann integrable on $[a,b]$, then the following holds.
\[\lim_{n\to \infty } \int^{b}_{a} f_n (x) dx =   \int^{b}_{a} \lim_{n \to \infty} f_n (x) dx \]
\end{theorem}
This result by Arzela is considerably difficult to prove for the Riemann integral, however the Lebesgue integral is an alternative definition of the integral that facilitates such proofs.

%- characteristic function
- 
- Arzela-ascoli theorem; Analogue of Bolzano-Weierstrass for sequences of real functions rather than sequences of real numbers.



\section{Weierstrass approximation theorem}

One extremely useful fact regarding. It assures one that continuous functions on a closed and bounded can be approximated to arbitrary precision by finding the right polynomial.

\begin{theorem}[Weierstrass approximation theorem]
	Let $f : I \to \mathbb{R}$ be a continuous real function on the closed and bounded (compact) interval $I$. For any $\varepsilon > 0$, there is a polynomial $p : I \to \mathbb{R}$ such that
\[ \sup (\{ |f(x)-p(x)| : x \in I\} )< \varepsilon\]
\end{theorem}

The Bernstein polynomials are used in numerical analysis to interpolate functions on closed bounded sets. Since they actually converge to their limit function uniformly, they also offer a constructive, elementary method to prove the Weierstrass approximation theorem.

Indeed, the \emph{Stone-Weierstrass theorem} uses techniques of functional analysis to consider other possible classes of functions that can uniformly approximate any continuous real function, as well as generalizing the type of function domain space for which the statement holds.


