# Question 21491: A Fixed Residue Gives an Infinite Strictly Increasing Constant Subsequence [Return to catalogue page 108](https://ocme.evemisslab.com/en/questions/page/108/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-foundations/#q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0041) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0041-21491 · q-20261014-advance-20261109-r02-bolzano-weierstrass-theorem-real-0041 ## Original question and givens Use the whole infinite real periodic sequence with nonempty period_values=[16, -17, 18/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=16≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate). - G001 — Use the whole infinite real periodic sequence with nonempty period_values=[16, -17, 18/2], starting at index0 and repeating that full block forever. This is a declared infinite law, not an observed finite prefix. Use residue=0 and subsequence parameter j=16≥0. Give (residue, j, k_j, value, constant_subsequence_certificate, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-bolzano-weierstrass-theorem-real](https://ocme.evemisslab.com/en/mko/mko-bolzano-weierstrass-theorem-real/) ## Step-by-step derivation S001 · impliesBefore source operation=bw-periodic-subsequence;original snapshot and mathematics retained After explicit ordered source output fields=[residue, j, k_j, value, constant_subsequence_certificate, certificate] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL period index=0;value=16;absolute value=16 After prefix absolute maximum after this original entry=16 Conditions: none Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates. Step identifiers and dependenciesRule: bw_every_period_entry_absolute_bound · Uses: G001, S001 S003 · impliesBefore ORIGINAL period index=1;value=-17;absolute value=17 After prefix absolute maximum after this original entry=17 Conditions: none Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates. Step identifiers and dependenciesRule: bw_every_period_entry_absolute_bound · Uses: G001, S002 S004 · impliesBefore ORIGINAL period index=2;value=9;absolute value=9 After prefix absolute maximum after this original entry=17 Conditions: none Scan absolute values over the full period rather than the maximum signed value.A negative entry may determine the bound;retain original positions and duplicates. Step identifiers and dependenciesRule: bw_every_period_entry_absolute_bound · Uses: G001, S003 S005 · impliesBefore ORIGINAL NONEMPTY full period=[16, -17, 9];period length=3 After for EVERY n>=0,unique n=3q+r with0<=r<3 gives a_n=period_values[r];therefore |a_n|<=17 Conditions: none Whole-sequence boundedness follows from the declared infinite periodic law and division with remainder,not observed future behavior.This proves the stated model,not the general Bolzano-Weierstrass theorem from finite samples. Step identifiers and dependenciesRule: bw_periodic_law_for_all_n_not_finite_observation · Uses: G001, S004 S006 · calculationBefore bw period times ORIGINAL j := mul(3, 16) After bw period times ORIGINAL j = 48 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore bw ORIGINAL residue subsequence evaluated index := sum(0, 48) After bw ORIGINAL residue subsequence evaluated index = 48 Conditions: none Substitute the original roles and established intermediates exactly,retaining order,signs and quantified conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL residue=0,period=3;k(j)=0+3j for EVERY j>=0;displayed j=16,k_j=48 After for ALL j,k(j+1)-k(j)=3>0;remainder always0;a_k(j)=16;for EVERY epsilon>0 error0