#Proof of equivalence (Big O-Notation)

9 messages · Page 1 of 1 (latest)

outer bear
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How do i prove that the following Definitions are equivalent for a given g: N -> R^(>0):

0(g) := {f: N -> R^(>0) | Ec ∈ R^(>0) : ( Vn ∈ N: (f(n) <= cg(n)))} and
0(g) := {f: N -> R^(>0) | Ec ∈ R^(>0) : (En0 ∈ N: ( Vn ∈ N: (n >= n0 => f(n) <= cg(n))))}

compact lodgeBOT
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pliant vault
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Take f(1) = 2, f(n) = 1/n for n > 1

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And g(n) = 1/n for any n > 0

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In the first definition, f is not O(g)

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But in the second one, it is

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So if anything you are most likely doomed to failure trying to prove that the two sets are equal