Consider the function
which is shown in Figure .
Clearly, *f*(*n*) is non-negative for all integers .
We wish to show that .
According to Definition ,
in order to show this we need to find an integer and a constant *c**>*0
such that for all integers , .

As with big oh, it does not matter what the particular constants are--as long as they exist!
E.g., suppose we choose *c*=1.
Then

Since for all values of , we conclude that .

So, we have that for *c*=1 and ,
for all integers .
Hence, .
Figure clearly shows
that the function is less than
the function *f*(*n*)=5*n*-64*n*+256 for all values of .
Of course, there are many other values of *c* and that will do.
For example, *c*=2 and .

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.