Subsequence

Mathematical statement

The map k:N→N must be strictly increasing, not merely injective, and b_j=a_{k(j)}. This slice uses affine k(j)=s+d*j with d≥1 as a symbolic all-j certificate. A finite increasing prefix is not an already-supplied infinite map; arbitrary rearrangements and finite samples are not automatic certificates.

Representative formula (exact TeX)

b_j=a_{k(j)}

Assumptions

• Index starts are nonnegative integers and index steps are positive integers.
• Arithmetic source sequences and index compositions use exact integer results.

Symbol scope

• k — strictly increasing index map
• b — subsequence
• a — source sequence
• j — sequence index

Intuition, definition, and misconceptions

Strict increase is stronger than arbitrary injectivity.

Composition of subsequences still preserves order.

A finite index prefix does not by itself supply a complete infinite index map.

Notation candidate: evaluate the strictly increasing index map at j before selecting term a_{k(j)}. Injectivity,a reordering or a finite increasing prefix alone does not replace that premise.

An arbitrary rearrangement or finite sample is not automatically a subsequence certificate.

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

• Sequence — The source and selected sequence are distinct objects.
• Injectivity and distinct inputs — Strict increase implies injectivity, not conversely.
• Usual order on the real numbers — The original index order must be preserved.

Verification boundary

The symbolic certificate here is limited to affine increasing indices; other maps need their own universal certificate. The statement explicitly uses k(j); alignment with the shared abbreviated representative subscript remains under review.

not_formalized

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

Next obligation

Add the Bolzano-Weierstrass theorem.

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.