Consider the function *f*(*n*)=8*n*+128 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 , .

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

Since (*n*+8)*>*0 for all values of ,
we conclude that .
I.e., .

So, we have that for *c*=1 and ,
for all integers .
Hence, .
Figure clearly shows
that the function is greater than
the function *f*(*n*)=8*n*+128 to the right of *n*=16.

Of course, there are many other values of *c* and that will do.
For example, *c*=2
and will do,
as will *c*=4 and .
(See Figure ).

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