Proving Big-$\Theta$ if and only if Big-$O$ and Big-$\Omega$

Question

Given the definitions of Big-$O$ and Big-$\Omega$, I'd like to prove that $f(n) = \Theta(g(n))$ if and only if $f(n) = O(g(n))$ and $f(n) = \Omega(g(n))$.

Here's what I've come up with, but I'm not sure it's a rigorous proof, or even correct. I'm looking for comments on both the structure and style of my proof as well as correctness.

Proof by contradiction.

Suppose $f(n) = \Theta(g(n))$ and either $f(n) \ne O(g(n))$ or $f(n) \ne \Omega(g(n))$.

By the definition of $\Theta$,

$0 \le c_1 g(n) \le f(n) \le c_2 g(n)$

For some positive constants $c_1$, $c_2$, and $n > n_0$.

But it was given that $f(n) \ne O(g(n))$, which means there exist no positive constants $c$, $n_0$ such that for all $n>n_0$,

$0 \le f(n) \le c \cdot g(n)$

Which is a contradiction. $\blacksquare$

Answer 1

To prove an if and only if statement, you must prove that $p \to q$ and $q\to p$. Applied to your statement, you must prove that: $$f(n) = \Theta(g(n)) \implies f(n) = \mathcal{O}(g(n)) \wedge f(n)=\Omega(g(n)) \tag{1}$$ ...and... $$f(n) = \mathcal{O}(g(n)) \wedge f(n)=\Omega(g(n))\implies f(n) = \Theta(g(n)) \tag{2}$$

Your proof only deals with the second case, and is therefore incomplete.

Also, I must ask, why did you go with a proof by contradiction for part 1? It appears that a direct proof would suffice:

Let $f$ and $g$ be functions, such that $f(n) = \Theta(g(n))$.

By definition of $\Theta$, there exist positive constants $k_1$ and $k_2$ such that, for sufficiently large $n$: $$k_1\cdot |g(n)| \le |f(n)| \le k_2\cdot |g(n)|$$

Thus, for sufficiently large $n$: $$|f(n)| \le k_2|g(n)|$$ Therefore $f(n) = \mathcal{O}(g(n))$ by definition.

And, for sufficiently large $n$: $$k_1|g(n)| \le |f(n)| $$ Therefore $f(n) = \Omega(g(n))$ by definition. $$\blacksquare$$

Still, part 2 needs to be proven for it to be an if and only if.