$f, g: ℝ+ → ℝ+.$ Prove or disprove: $f(n) + g(n) ∈ O(g(n)f(n))$
My try: Let us assume the theorem holds true. By definition, we get that: $$⇒ f(n) + g(n) ≤ cg(n)f(n)$$ dividing both sides by $g(n)f(n)$:
$$\frac{1}{f\left(n\right)}+\frac{1}{g\left(n\right)}\le c$$ Now, consider $g(n) = f(n) = 0.5^n$ with the domain of $n > 0$. Subsituting: $$\frac{1}{0.5^n}+\frac{1}{0.5^n}\le c$$ Now, take the $\lim _{n\to \infty }$ of both sides, we get that: $$\infty \le c$$ and this is a contradiction since infinity is bigger than any real number. And thus, our initial assumption that $f(n)+g(n)∈O(g(n)f(n))$ is false. Is my proof valid?