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We have not n2 = O(n).
Proof: We suppose n2 = O(n) and show a contradiction. By the assumption, we have some c in R and m in N with
If c < 0, let k:=max(1, m); then we have k >= m but
forall n >= m: |n2| <= c|n|.
If c >= 0, let k := max(m, ceiling(c+2)). Then we have k >= m but
|k2| > 0 > c|k|
|k2| = k2 >= ceiling(c+2)2 >= (c+2)2 = c2 +4c + 4 > c2 +3c = c(c+3) >= cceiling(c+2) >= ck = c|k|