Chapter 4, Problem 18R

Explanation of Solution

Given:

It is given that if $d\left(n\right)$ is $O\left(f\left(n\right)\right)$ and $f\left(n\right)$ is $O\left(g\left(n\right)\right)$ then $d\left(n\right)$ is $O\left(g\left(n\right)\right)$.

Big-Oh notation:

In big-Oh notation, let “f” and “g” be functions from the integers or the real numbers to the real numbers. It means that $\text{f}\left(\text{x}\right)$ is “$\text{O}\left(\text{g}\left(\text{x}\right)\right)$” if there are constants “C” and “k” such that

$\left|\text{f}\left(\text{x}\right)\right|\le \text{C}\left|\text{g}\left(\text{x}\right)\right|\text{ where }\left(\text{x > k}\right)$

Proof:

Let us assume that $g\left(n\right)$ and $f\left(n\right)$ be functions mapping non-negative integers to real numbers