# Arbitrarily anti Cauchy

I made a pact with myself to refrain from doing any silly fun questions until I’d done at least a week of serious study. It seems I have failed. Maybe a slower taper is in order here… anyway:

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Note: Here all sequences are indexed by the naturals.

Call a real-valued sequence a_n anti-Cauchy if for any E > 0, there exists a P such that

|a_i – a_j |> E for all distinct i, j > P

Given any sequence a_n, for each natural N >= 0, define the N’th difference sequence of a_n, {D(N)_i} (the name of the sequence is D(N), it is indexed by i) as follows:

D(0)_i = a_i

D(N+1)_i = D(N)(i+1) – D(N)_i

Prove or disprove: For every subset S of the natural numbers, there exists a sequence such that it’s k’th difference sequence is anti-Cauchy exactly when k is in S.

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Someone on this sub gave this to me as an extension to one of my problems. He seemed to imply it was true, but I haven’t been able to figure it out and it’s been bugging me ever since.

submitted by /u/WaltWhit3