# Sequence

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

This entry uses real sequences a:N→R indexed from 0. Sequence terms are ordered by their indices and cannot be rearranged into an unordered set.

mko-sequence · definition · v0.10.0



## Mathematical statement

Under the convention N={0,1,2,…}, a real sequence is a function a:N→R whose nth term is written a(n) or a_n. A finite sequence instead states a length L and the index set {0,…,L-1}.





## Representative formula (exact TeX)

a:N\to R





## Assumptions


- An infinite sequence uses N, the natural numbers including 0, and calls a_0 the initial term.
- The length L of a finite sequence is a nonnegative integer and only 0≤n<L is permitted; a recurrence includes initial conditions.





## Symbol scope


- a — real sequence
- N — natural-number indices including 0
- R — real codomain





## Intuition, definition, and misconceptions

The same value may occur at different indices of a sequence. Index order is part of the function data, unlike an unordered set.

An explicit formula directly specifies a_n. A recurrence uses initial conditions and earlier terms to specify later terms. Finite computations here use the explicit rule a_n=u+dn or the recurrence a_0=u and a_(n+1)=a_n+d.

A few initial terms usually do not determine a unique continuation rule. Sequence limits, monotonicity, infinite series, and general existence and uniqueness of recurrences remain later obligations.

Common mistakes include confusing zero-based and one-based indexing or claiming that one finite prefix determines a unique sequence.





## Adopted derivation guide

[Sequences: fixing the general term from an initial condition and a recurrence, and the pitfalls of indexing and finite prefixes](https://ocme.evemisslab.com/en/derivations/derivation-mko-sequence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A sequence assigns one term to each index.
- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — The natural-number indices begin at 0.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — The codomain of the sequences in this entry is restricted to the real numbers.





## Verification boundary

Finite terms under a specified formula or recurrence can be recomputed exactly. No claim is made that the sequence converges or that an infinite property has been proved.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Bind Nat→Real, finite indices Fin L, and recurrences with explicit initial values to exact semantics.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/sequence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-sequence/index.md)