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Sequences and Limit Theorems

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Sequences  ::  Convergence  ::  Limit  ::  Central Limit

Convergent sequences of real numbers and functions


\begin{definition} Let $x_1,x_2,\ldots$ be a sequence of real numbers. This se... ...e $x_n \rightarrow x$, or $\lim_{n\rightarrow \infty} x_n = x$. \end{definition}
For real numbers (which are complete), a necessary and sufficient condition:

\begin{displaymath}\{x_n\}_{n=1}^\infty \text{ converges} \Leftrightarrow \lim_{n\rightarrow\infty} \sup_{m>n} \vert x_m - x_n\vert = 0. \end{displaymath}n} \vert x_m - x_n\vert = 0. \end{displaymath}" border="0" height="49" width="317" />

The latter condition says that $\{x_n\}$ is a Cauchy sequence.
\begin{definition} Suppose $f_1, f_2, \ldots$ is a sequence of {\em functions}... ...hat $\vert f_n(x) - f(x)\vert < \epsilon$ for all $n \geq N$. \end{definition}< \epsilon$ for all $n \geq N$. \end{definition}" align="bottom" border="0" height="54" width="555" />
(It may be necessary to choose a different $N$ for each $x$.)
\begin{definition} We say that $f_n$ converges {\bf uniformly} to $f$ if for ... ...psilon$ for all $n \geq N$ {\bf and for all } $x \in \Omega$. \end{definition}
Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, May 31). Sequences and Limit Theorems. Retrieved November 23, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lec5_1.html. This work is licensed under a Creative Commons License. Creative Commons License
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