Topology MOC

Convergence

A sequence (𝑎𝑛)∞𝑛=1 in a topological space (𝑋,T) is converges to 𝑎¹ iff for every open neighbourhood 𝑈 of 𝑎, there exists an integer 𝑁 such that 𝑎𝑛 ∈𝑈 for all 𝑛 >𝑁. #m/def/topology We then write

(𝑎𝑛)→𝑎

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 (𝑎𝑛)∞𝑛=1 tends towards a limit 𝐿 iff. for every 𝜖 >0 there exists an integer 𝑁 such that |𝑎𝑛−𝐿| <𝜖 for all 𝑛 >𝑁. #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 (𝑋,𝑑)

A sequence (𝑎𝑛)∞𝑛=1 in 𝑋 tends towards a point 𝐿 ∈𝑋 iff. for every 𝜖 >0 there exists an integer 𝑁 such that 𝑑(𝑎𝑛,𝐿) <𝜖 for all 𝑛 >𝑁, i.e. 𝑎𝑛 ∈𝐵𝜖(𝐿) 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 𝛼 >0

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

2.

\begin{align*} \lim_{ n \to \infty } \sqrt[n]{n} = \lim_{ n \to \infty } n^{1/n} = 1 \end{align*}

3.𝐹𝑜𝑟𝑎𝑛𝑦$𝑎∈ℝ$

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

4.𝐹𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡$𝑎∈ℝ$

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

[^2022]: 2022\. [[Sources/@bassomMATH1012MathematicalTheory2022|MATH1012: Mathematical theory and methods]], Theorem 8.2.1, p. 119 # --- #state/tidy | #SemBr

Footnotes

1. German konvergiert gegen 𝑎 ↩