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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|.
by induction on n and thus have a contradiction.
forall n >= 0: n2 <= 2*2n
See lecture notes.