As a homework assignment I am trying to prove/disprove the next statement:
Let $f(x)=O_a(g(x))$, then $\forall A,B\in\mathbb{R}\rightarrow A\cdot f(x)=O_a(B \cdot g(x))$
Which I think is wrong and thus trying to disprove.
So assuming I'm right, my question is logical: we know that there exist a $C_1 \gt 0$ such that $\|f(x)\| \le C_1 \cdot \|g(X)\|$ for all $x$ in a neighborhood of $a$. In order to disprove the given statement, should I show that there exist $A,B\in\mathbb{R}$ such that $\|f(x)\| \gt C_1 \cdot \|g(X)\|$ for ALL $C \gt 0$, or for a specific $C_1$?
Hope I was clear, thanks