Sequence

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — A sequence assigns one term to each index.
• Natural number — The natural-number indices begin at 0.
• 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.