Choosing c and $n_{0}$ for big-oh family definitions

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As far as i know the definitions are:

$f(n)$ is $O(g(n))$ iff there exists $ c > 0, n_{0} $ such that $f(n) \le c \times g(n)$ holds for all $n \ge n_{0}$

$f(n)$ is $o(g(n))$ iff there exists $ c, n_{0} $ such that $f(n) \lt c \times g(n)$ holds for all $ c > 0 $ and $n \ge n_{0}$, where $n$ can depend on $c$, for example $n_{0} = c /2$

$f(n)$ is $\theta(g(n))$ iff there exists $ c', c'' > 0, n_{0} $ such that $f(n) \le c' \times g(n)$ and $f(n) \ge c'' \times g(n)$ holds for all $n \ge n_{0}$

$f(n)$ is $\Omega(g(n))$ iff there exists $ c > 0, n_{0}$, such that $f(n) \ge c \times g(n)$ holds for all $n \ge n_{0}$

The part that I am not sure about is the correct ranges to choose $c, n$ from. Are the definitions I wrote correct?