Go backward to ExampleGo up to TopGo forward to O Manipulation |

We have `n`^{2} = O(2^{n}):

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

forallcinR,minN: ~foralln>=m: |n^{2}| <=c|2^{n}|.

We prove

by induction on

foralln>= 0:n^{2}<= 2*2^{n}

*See lecture notes.*

Author: Wolfgang Schreiner

Last Modification: December 14, 1999