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Example

We have n² = O(2ⁿ):

Proof: We assume not n² = O(2ⁿ) and show a contradiction. By assumption, we have ~exists c in R, m in N: forall n >= m: |n²| <= c|2ⁿ|, i.e.,

forall c in R, m in N: ~forall n >= m: |n2| <= c|2n|.

We prove

forall n >= 0: n2 <= 2*2n

See lecture notes.