Textbook Problem

Prove Theorem 11.2.9:
a. Let $f_1,f_2$ , and g be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that $f_1\left(n\right)\ge 0,f_2\left(n\right)\ge 0$ , and $g\left(n\right)\ge 0$ for every integer $n\ge r$. If $f_1\left(n\right)$is $\Theta \left(g\left(n\right)\right)$and $f_2\left(n\right)$is $\Theta \left(g\left(n\right)\right)$, then $\left(f_1\left(n\right)+f_2\left(n\right)\right)$is $\Theta \left(g\left(n\right)\right)$ .

b. Let $f_1,f_2,g_1$ , and $g_2$ , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that $f_1\left(n\right)\ge 0,f_2\left(n\right)\ge 0,g_1\left(n\right)\ge 0$ , and $g_2\left(n\right)\ge 0$ for every integer $n\ge r$. If $f_1\left(n\right)$ is $\Theta \left(g_1\left(n\right)\right)$ and $f_2\left(n\right)$ is $\Theta \left(g_2\left(n\right)\right)$, then $\left(f_1\left(n\right)f_2\left(n\right)\right)$is $\Theta \left(g_1\left(n\right)g_2\left(n\right)\right)$ .

c. Let $f_1,f_2,g_1$ , and $g_2$ , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that $f_1\left(n\right)\ge 0,f_2\left(n\right)\ge 0,g_1\left(n\right)\ge 0$ , and $g_2\left(n\right)\ge 0$ for every integer $n\ge r$. If $f_1\left(n\right)$ is $\Theta \left(g_1\left(n\right)\right)$ and $f_2\left(n\right)$is $\Theta \left(g_2\left(n\right)\right)$and if there is a real number s so that $g_1\left(n\right)\le g_2\left(n\right)$for every integer $n\ge s$ , then $\left(f_1\left(n\right)+f_2\left(n\right)\right)$ is $\Theta \left(g_2\left(n\right)\right)$ .