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\begin{document}

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\section*{convergence}

\begin{quote}%
This entry is about the notion of limit in [[analysis]] and [[topology]]. For the notion of the same name in [[category theory]] see at \emph{[[limit]]}.


\end{quote}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{notation}{Notation}\dotfill \pageref*{notation} \linebreak
\noindent\hyperlink{limits_of_filters}{Limits of filters}\dotfill \pageref*{limits_of_filters} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToLimitsInCategoryTheory}{Relation to limits in the sense of category theory}\dotfill \pageref*{RelationToLimitsInCategoryTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{limit} of a [[sequence]] (or [[net]]) of points $(x_i)$ in a [[topological space]] (or other [[convergence space]]) $X$ is a point $x$ such that the sequence eventually gets arbitrarily close to $x$. We can also speak of a limit of a [[filter]] on $X$.

The notion is of particular and historical importance in [[analysis]], where it serves to define for instance the notion of [[derivative]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

The precise definition depends on what sort of space $X$ is.

\begin{defn}
\label{topological}\hypertarget{topological}{}
If $X$ a [[topological space]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of points in $X$, one says that a point $x \in X$ is a \textbf{limit} of $\nu$ or that $\nu$ \textbf{converges} to $x$ if for each [[neighbourhood]] $U$ in $X$ of $x$ there exists an $n \in I$ such that $x_i \in U$ for each $i \geq n$.

\end{defn}
An important special case (the original) is:

\begin{defn}
\label{real}\hypertarget{real}{}
If $X$ the [[real line]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of [[real numbers]], one says that a point $x \in X$ is a \textbf{limit} of $\nu$ or that $\nu$ \textbf{converges} to $x$ if for each [[positive number]] $\epsilon$ there exists an $n \in I$ such that ${|x_i - x|} \lt \epsilon$ for each $i \geq n$.

\end{defn}
An important generalization (possibly the most general) is:

\begin{defn}
\label{convergence}\hypertarget{convergence}{}
If $X$ a [[convergence space]] and $I$ the set of [[natural numbers]] (or more generally any [[directed set]]) and $\nu = (x_i)_{i\in I}\colon I \to X$ is a [[sequence]] (or a [[net]]) of points in $X$, one says that a point $x \in X$ is a \textbf{limit} of $\nu$ or that $\nu$ \textbf{converges} to $x$ if the [[eventuality filter]] of $\nu$ converges to $x$ (which is a primitive concept in convergence spaces).

\end{defn}
Other types of [[space]] for which we might put in definitions (or which might have definitions on their own pages) are ([[extended metric space|extended]]) ([[quasimetric space|quasi]])-([[pseudometric space|pseudo]])-[[metric spaces]], ([[quasiuniform space|quasi]])-[[uniform spaces]], [[pretopological spaces]], and (quasi)-[[uniform convergence spaces]].

\hypertarget{notation}{}\subsubsection*{{Notation}}\label{notation}

When one of the conditions above holds, we may write any of the following, where `$\to$' is read as `converges to':

\begin{itemize}%
\item $\nu \to x$,
\item $x_i \to_i x$,
\item $x_i \to x$ (suppressing the index $i$ by abuse of notation).

\end{itemize}
Or we may write any of the following, were $\lim$ is read as `the set of limits of':

\begin{itemize}%
\item $x \in \lim \nu$,
\item $x \in \lim_i x_i$,
\item $x \in \lim x_i$ (suppressing the index $i$ by abuse of notation).

\end{itemize}
Of course, the right-hand side has a meaning by itself, as the set of limits itself (a [[subset]] of the [[underlying set]] of $X$, or a [[subspace]] of $X$ itself).

If $X$ is a [[Hausdorff space]], then there is at most one point $x$ with the property that the sequence (or net) $\nu$ converges to $x$. Then we may write any of the following, were now $\lim$ is read as `the limit of':

\begin{itemize}%
\item $x = \lim \nu$,
\item $x = \lim_i x_i$,
\item $x = \lim x_i$ (suppressing the index $i$ by abuse of notation).

\end{itemize}
Now the right-hand side by itself is the possibly undefined [[term]] for the limit itself (if it exists).

\hypertarget{limits_of_filters}{}\subsubsection*{{Limits of filters}}\label{limits_of_filters}

More generally than [[sequences]], and equivalently to [[nets]], we may speak of limits of [[filters]] on $X$. This concept is axiomatized directly in the concept of [[convergence space]]. In the case of a [[topological space]] $X$, a filter of subsets of $X$ converges to a point $x$ if every neighbourhood of $x$ is contained in the filter.

In the definitions above, equivalent nets (those with equal [[eventuality filters]]) always converge to the same point. As every [[proper filter]] is the eventuality filter of some net, a proper filter converges to $x$ if any of these nets converges to $x$; the [[improper filter]] converges to every point. (In [[constructive mathematics]], we may cover all filters by saying: $F$ converges to $x$ if, on the assumption that $F$ is proper, any of its nets converges to $x$.)

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToLimitsInCategoryTheory}{}\subsubsection*{{Relation to limits in the sense of category theory}}\label{RelationToLimitsInCategoryTheory}

The [[limits]] of [[category theory]] are a great generalization of an analogy with the limits discussed here. It turns out, however, that limits in [[topological spaces]] (at least) can be viewed as category-theoretic limits. For now, see \href{http://math.stackexchange.com/a/62800}{this math.sx answer}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[geometric series]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[filter]]


\item [[convergence space]]


\item [[series]]


\item [[limit point]]


\item [[radius of convergence]]


\item [[uniform convergence]], [[pointwise convergence]]


\item [[divergent series]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Limit_%28mathematics%29}{Limit (mathematics)}}

\end{itemize}
Discussion of this history of the concept, with emphasis on its roots all the way back in [[Zeno's paradoxes of motion]] is in

\begin{itemize}%
\item Carl Benjamin Boyer, \emph{The history of the Calculus and its conceptual development}, Dover 1949

\end{itemize}
category: analysis

[[!redirects convergence]] [[!redirects convergences]] [[!redirects convergencies]]

[[!redirects converges]]

[[!redirects convergence of a sequence]] [[!redirects convergence of sequences]] [[!redirects convergence of a net]] [[!redirects convergence of nets]] [[!redirects convergence of a filter]] [[!redirects convergence of filters]]

[[!redirects convergent sequence]] [[!redirects convergent sequences]] [[!redirects convergent net]] [[!redirects convergent nets]] [[!redirects convergent filter]] [[!redirects convergent filters]]

[[!redirects limit of a sequence]] [[!redirects limits of a sequence]] [[!redirects limits of sequences]] [[!redirects limit of a net]] [[!redirects limits of a net]] [[!redirects limits of nets]] [[!redirects limit of a filter]] [[!redirects limits of a filter]] [[!redirects limits of filters]]



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