Data Structures and Algorithms
with Object-Oriented Design Patterns in Java
Consider the function f(n)=8n+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! For example, suppose we choose c=1. Then
Since (n+8)>0 for all values of ,
we conclude that 
.
That is, 
.
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)=8n+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 © 1998 by Bruno R. Preiss, P.Eng. All rights reserved.
.