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We have *not* `n`^{2} =
O(`n`).

*Proof:* We suppose
`n`^{2} = O(`n`) and show a contradiction.
By the assumption, we have some `c` in **R** and `m`
in **N** with

If

foralln>=m: |n^{2}| <=c|n|.

If

| k^{2}| > 0 >c|k|

| k^{2}| =k^{2}>= ceiling(c+2)^{2}>= (c+2)^{2}=c^{2}+4c+ 4> c^{2}+3c=c(c+3) >=cceiling(c+2) >=ck=c|k|

Author: Wolfgang Schreiner

Last Modification: December 14, 1999