I am wondering how to notate "for all positive real value $c$"
Is there a correct notation among the following?
$$
\forall c \in \mathbb{R} > 0\\
\forall c \left( \in \mathbb{R} \right) > 0\\
\forall c > 0 \in \mathbb{R}\\
\forall c > 0 \left( \in \mathbb{R} \right)\\
$$
My ultimate goal is notating the following sentence.
"$o(g(n))=\{f(n):$ For any constant positive real value $c$, there is a constant $n_0$ such that $0 \le f(n) \lt cg(n)$ for all $n \ge n_0\}$"
My trial is
$$
o(g(n))=\{f(n):\forall c>0(c\in\mathbb R), \exists n_0\in\mathbb{N} \ \ \ \ s.t.\ \forall n>n_0,\ \ 0 \le f(n) \lt cg(n)\}
$$
I want to correct this part: $\forall c>0(c\in\mathbb R)$
Best Answer
If you would like to have mercy on your reader, please avoid squeezing too many relations together. "$\forall c \in \mathbb{R} > 0$", for example, is readable, but it is not logically precise.
There are at least two ways out; the first one is to say "for all $c \in \mathbb{R}$ such that $c > 0$", and the other is to define the set of all reals $> 0$ and say "for all $c$ in the set ".
You may also use "for all positive $c \in \mathbb{R}$", but this is risky if you do not specify in the first place what your "positive" means; for people may interpret "positive" differently.
In sum, the precise and safe way seems to be "for all $c \in \mathbb{R}$ such that $c > 0$".