$g ( n ) ∈ ω ( 1 )$ and $f ( n ) ∈ o ( g ( n ) )$ imply $2 f ( n ) ∈ o ( 2 g ( n ) )$
Considering non negative case we have
$$O(f)=\{g: \exists C&gt;0, \exists N \in \mathbb{N}, \forall n&gt;N, g(n) \leqslant Cf(n)\}$$
From this definition for $\forall C&gt;0,$ we have:
$$ C \cdot O(f)= O(C \cdot f) = O(f)$$
It can be written also as:
$$C \cdot g \in O(f) \Leftrightarrow g \in O(C \cdot f) \Leftrightarrow g \in O(f)$$
without any additional condition.