Two convergent subsequences of an oscillatory tail

Goal and scope

For a_n=(-1)^n+1/(n+1), provide boundedness and two complete convergent subsequences.

Only the declared symbolic sequence, not a replacement for the general BW theorem.

Givens and symbols

• algebra
• epsilon

• G1 — a_n=(-1)^n+1/(n+1), n≥0; take k(j)=2j and ell(j)=2j+1.

• k,\ell — parity indices
• M — whole-sequence bound

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — Boundedness means there exists M bounding all terms, not just a finite prefix. A subsequence has an infinite strictly increasing index map, and convergence has an all-later epsilon-tail witness. Periodic sequences and symbolic oscillatory rational tails give exact instances here. A finite prefix or finite cluster search is insufficient. The theorem does not require the original sequence to converge or guarantee that every convergent subsequence has the same limit. · mko-bolzano-weierstrass-theorem-real
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals.

Main derivation

|(-1)^n|=1,\quad0<1/(n+1)\le1

|a_n|\le2\quad\forall n\ge0

k(j+1)-k(j)=2,\quad\ell(j+1)-\ell(j)=2

k,\ell:\mathbb N_0\to\mathbb N_0\text{ strictly increasing}

a_{k(j)}=a_{2j}

a_{2j}=1+1/(2j+1)

\varepsilon>0,\quad J=\lfloor1/\varepsilon\rfloor+1

j\ge J\Longrightarrow |a_{2j}-1|=1/(2j+1)<\varepsilon

a_{\ell(j)}=a_{2j+1}

a_{2j+1}=-1+1/(2j+2)

j\ge J=\lfloor1/\varepsilon\rfloor+1

|a_{2j+1}+1|=1/(2j+2)<\varepsilon

a_{2j}\to1,\quad a_{2j+1}\to-1

(a_n)\text{ has convergent subsequences with different limits}

1\ne-1

(a_n)\text{ does not converge}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S04, S06 — Nonconvergence follows from different proved subsequential limits, not visual oscillation.
• S07 · premise_audit · S01, S02, S04, S06 — The instance does not claim the theorem for all bounded sequences from finite tests.
• S06 · premise_audit · S05 — This independently checks the odd subsequence, rather than inferring the even limit for it.
• S05 · inverse_equivalence · G1, S02 — The limit is -1, not its magnitude 1.
• S04 · premise_audit · S03 — The strict bound holds for every j≥J.
• S03 · inverse_equivalence · G1, S02 — The denominator is 2j+1, not 2j.
• S02 · premise_audit · G1 — The infinite index domain is specified, not a finite list.
• S01 · premise_audit · G1 — A prefix maximum is not substituted for whole-sequence boundedness.

Error controls and negative controls

Whole-sequence bound

M must hold for every n.

Three subsequence obligations

Indices must be strictly increasing and infinite, with an all-tail convergence bound.

• BW guarantees original-sequence convergence. → reject — This example has two different proved subsequential limits.
• All convergent subsequences must have the same limit. → reject — The limits 1 and -1 here differ.
• Finite repeated values prove BW. → reject — Complete infinite indices, a whole-sequence bound and an all-tail certificate are required.

Result and limitations

The whole sequence is bounded; the even subsequence tends to 1 and the odd to -1, so the original does not converge.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: f1198b016a1e24a4b4724094afffa9b2aa9f46a46781515f4f078244bea62398
• original source SHA-256: 605ab67bfeb763ae5cddd56fa58c8fa84b2fbce158ebd93195090c891bdf3c42
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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