if f and g are monotonically increasing functions, such that f(g(n))=O(n) and f(n)=Ω(n) then g(n)=O(n)

Question

I have to prove this statement : if $f$ and $g$ are monotonically increasing functions, such that $f(g(n))=O(n)$ and $f(n)=Ω(n)$ then $g(n)=O(n).$

Answer 1

By truncating the functions we can assume that the starting indices are $0$. That is, $$ \exists c,c_0 > 0\mid \forall n \in \mathbb{W}, f(g(n)) \leq c n, f(n) \geq c_0 n $$ Then we have $$ c n \geq f(g(n)) \geq c_0 g(n) $$ as desired.