Sequences  

In General
$ Def: A sequence in a set X is a subset AX and a map \(\mathbb N\) → A, n \(\mapsto\) an, which is onto, so that A = {an | n ∈ \(\mathbb N\)}.
> Online resources: The On-Line Encyclopedia of Integer Sequences (OEIS) site; Wikipedia page.

Operations on Sequences
$ Difference operator: The mapping {an} → {bn} defined by bn = an+1an, for all n.

Types of Sequences
$ Cauchy sequence: A sequence {an} in a normed space such that, for all ε > 0, ∃ N such that for all n, m > N, || anam|| < ε.
$ Monotonic sequence: (Increasing) One such that the difference operator produces a non-negative sequence.
$ Completely monotonic sequence: (Increasing) One such that k iterations of the difference operator produce a non-negative sequence, for all k.

Convergence > s.a. limit; series.
$ In a topological space (X, T): A sequence {xn} converges to x0 iff for all U T, x0U, ∃ N such that n > N implies xnU.
$ In a metric space (X, d): A sequence {xn} converges to x0 iff for all ε > 0, ∃ N such that n > N implies d(xn, x0) < ε.
$ In a normed vector space: The sequence {xn} converges strongly s to x0 iff || xnx0|| → 0 as n → ∞.
$ For random variables: A sequence {xn} of random variables in a metric space (X, d) converges in probability to a random variable x if \(\forall\epsilon,\,\lim_{n\to\infty}P(d(x_n,x) > \epsilon) = 0\) (this is stronger than convergence in distribution).

Fibonacci Sequence > s.a. Golden Ratio.
$ Def: The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ..., in which Fn = Fn–2 + Fn–1; It appears in life patterns.
* Result: (Zeckendorf) Every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers.
@ General references: Vorobiev 02 [IIb]; Posamentier & Lehmann 07; Miller & Wang JCTA(12) [and central-limit-type theorems].
@ In nature: news pw(07)apr [stress-related Fibonacci spirals]; Pennybacker & Newell PRL(13) [underlying physical and biochemical mechanisms in plant growth].
> Online resources: See Wikipedia page.

Other Examples > see Wikipedia page on integer sequences.

Generalizations > s.a. projective family.
* Inductive family or system: Given a directed set I, an inductive family on I is a collection {Xi | iI} of objects in a category, and immersions {φij: XiXi | ijI}, such that φii = idXi, and φij \(\circ\) φjk = φik; Remark: In many (all?) categories each such family defines an inductive limit.


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