Exercise 3.1.1

Let $f(n)$ + $g(n)$ be asymptotically nonnegative functions. Using the basic definition of $\Theta$-notation, prove that $\max(f(n), g(n)) = \Theta(f(n) + g(n))$.

From "asymptotically nonnegative", we can assume that

$$\begin{align} \exists n_1, n_2: & f(n) \geq 0 & \text{for } n > n_1 \\ & g(n) \geq 0 & \text{for } n > n_2 \end{align}$$

Let $n_0 = max(n_1, n_2)$. Some obvious things for $n > n_0$:

$$ f(n) \leq \max(f(n), g(n)) \\ g(n) \leq \max(f(n), g(n)) \\ \big(f(n) + g(n)\big)/2 \leq \max(f(n), g(n)) \\ \max(f(n), g(n)) \leq f(n) + g(n) $$

From the last two inequalities, we get:

$$ 0 \leq \frac{1}{2}\big(f(n) + g(n)\big) \leq \min\big(f(n), g(n)\big) \leq f(n) + g(n) \quad \text{for } n > n_0 $$

Which is the definition of $\Theta(f(n) + g(n))$ with $c_1 = 1/2, c_2 = 1$.