@DanielFischer Nice.. I am looking at this exercise: Prove or disapprove the sentence $f(n) \in O(f(n)^2)$.
Could I answer as followed?
False. Let $f(n)=\frac{1}{n}$, $\forall n \in \mathbb{N}$. Suppose that $f(n)=O(f(n)^2)$. That means that there are $c>0$ and $n_0 \in \mathbb{N}$ such that $ \frac{1}{n}=f(n) \leq c f(n)^2=c \frac{1}{n^2}, \forall n \geq n_0$.
$\frac{1}{n} \leq c \frac{1}{n^2} \Rightarrow \frac{n^2}{n} \leq c \Rightarrow n \geq c$, contradiction.
We see that $f(n) \in O(f(n)^2)$ iff $f(n)=\Omega(1)$.