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Monday, February 13, 2006

monotonic subsequence : example of proof by cases

Proof By Cases: Example from monotonic subsequence theorem
tag: math. proof. case.

"every infinite sequence of real numbers have a monotonic infinite subsequence."

the conclusion has two CASES: having nonincreasing infinite subsequence and having nondecreasing infinite subsequence.

one proof (in Zakon series) starts by dividing two CASES:
case 1: SOME subsequence of the sequence don't have maximum value.
case 2: ALL subsequence of the sequence have max.
SOME or ALL

another proof (in everything2) starts by dividing cases:
case 1: infinitely many maximal index.
case 2: only finitely many maximal index. so that no index is a maximal index from on some N.
, where maximal index is defined as http://everything2.com/index.pl?node_id=983077
Infinite occurence or only finitely many occurence.

another proof starts by dividing:
case 1: unbounded sequence
case 2: bounded sequence.
, then case 2 is more divided into cases,
, as in http://www.puc.edu/Faculty/George_Hilton/id94.htm
unbounded or bounded.

all the three are proofs by cases.

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