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: $\forall c>0(c\in\mathbb R)$