If $f:\mathbb{C}\to \mathbb{C}$ is a function and $$-\Im(f(x+iy))=\mathcal{O}(y)$$ where $\mathcal{O}$ is the big O notation, then prove that $$\Im(f(x+iy))=\mathcal{O}(y) \ \ \ \ \text{as} \ \ \ \ y\to \infty$$
I tried: Since $$-\Im(f(x+iy))=\mathcal{O}(y) \ \ \ \ \text{as} \ \ \ \ y\to \infty $$ so there exists $M,y_0$ such that $$|\Im(f(x+iy))|\leq M y $$ whenever $y\geq y_0$.
So can we say from the above that $$\Im(f(x+iy))=\mathcal{O}(y) \ \ \ \ \text{as} \ \ \ \ y\to \infty $$
Do we require the limit definition of big O?