Topology MOC

Convergence

A sequence in a topological space is converges to ¹ iff for every open neighbourhood of , there exists an integer such that for all . #m/def/topology We then write

Note that the uniqueness of the limit is only guaranteed if the space is Hausdorff (in the Trivial topology every sequence converges to every point).

In particular spaces

Real numbers

Using a construction analogous to the Epsilon-Delta Construction of the Limit,

A real sequence tends towards a limit iff. for every there exists an integer such that for all . #m/def/calculus

The familiar language and properties of Limits (Calculus) applies — particularly useful may be the Limit Laws.

Metric space

This can be extended to any metric space

A sequence in tends towards a point iff. for every there exists an integer such that for all , i.e. using the concept of an Open ball. #m/def/anal

This definition is useful for defining the Limits in a function space. The concept of convergence in a metric space is generalised to the Cauchy sequence.

Particular real limits

The following limits are particularly useful[^2022]

1. For

\begin{align*} \lim_{ n \to \infty } \frac{\ln n}{n^\alpha} = 0 \end{align*}

3. For any

\begin{align*} \lim_{ n \to \infty } \frac{a^n}{n!}=0 \end{align*}

\begin{align*} \lim_{ n \to \infty } \left( 1 + \frac{a}{n} \right)^n = e^a \end{align*}

Footnotes

1. German konvergiert gegen ↩