Intuition behind $o(n)+\omega(n)+\Theta(n)=\Omega(n)$

Question

Intuition behind $o(n)+\omega(n)+\Theta(n)=\Omega(n)$

Left hand side means set of all functions more than n + set of all functions greater than n + set of all functions equal to n ---> what does it mean? how is it equal to set of all functions greater than or equal to n. How to think about it?

Answer 1

Check definitions:

$o(n)$ is a function $f_1(n)$ so that $\lim_{n \to \infty} f_1(n) / n = 0$

$\omega(n)$ is a function $f_2(n)$ so that $\lim_{n \to \infty} f_2(n) / n = \infty$

$\Theta(n)$ is a function $f_3(n)$ so that for some constants $c n \le f_3(n) \le C n$

$\Omega(n)$ is some function $f_4(n)$ that grows possibly faster than $c n$ for some $c$

From the above it is intuitive.

For a formal proof, set up the $\epsilon$ -- $N_0$ as the definition states for the various limits and grind through it.