Convergence#

Definition: Convergence of Sequences

Let \((X, \tau)\) be a topological space and let \((x_n)_{n \in \mathcal{D}}\) be a sequence of points in \(X\).

We say that \((x_n)_{n \in \mathcal{D}}\) is convergent if there exists some \(L \in X\) such that for each neighborhood \(N(L)\) there exists some \(N_0 \in I\) with

\[x_n \in N(L) \qquad \forall n \ge N_0.\]

Definition: Limit

Any such \(L\) is called a limit of \((x_n)_{n \in \mathcal{D}}\). We also say that \((x_n)_{n \in \mathcal{D}}\) converges to \(L\).

Notation

\[ \lim_{n \to \infty} x_n = L \qquad x_n \to L \]

Warning: Non-Uniqueness of Limits

In general, \((x_n)_{n \in \mathcal{D}}\) may have multiple different limits.

Limit Inferior and Limit Superior#

Definition: Limit Inferior

Let \((X, \tau)\) be a topological space such that \(X\) is partially ordered and let \((x_n)_{n \in \mathcal{D}}\) be a sequence of points in \(X\).

The limit inferior of \((x_n)_{n \in \mathcal{D}}\) is the supremum of the infima of all tail sequences of \((x_n)_{n \in \mathcal{D}}\):

\[ \sup_{n \in I} \inf \{ x_k \mid k \geq n \} \]

Notation

If the limit inferior exists, we denote it in the following way:

\[ \liminf_{n \to \infty} x_n \]

Definition: Limit Superior

Let \((X, \tau)\) be a topological space such that \(X\) is partially ordered and let \((x_n)_{n \in \mathcal{D}}\) be a sequence of points in \(X\).

The limit superior of \((x_n)_{n \in \mathcal{D}}\) is the infimum of the suprema of all tail sequences of \((x_n)_{n \in \mathcal{D}}\):

\[ \inf_{n \in I} \sup \{ x_k \mid k \geq n \} \]

Notation

If the limit superior exists, we denote it in the following way:

\[ \limsup_{n \to \infty} x_n \]