if $f(n)$ is defined in the domain [0,1] then what $f(n) = O(1/n)$ actually means?
In this context it is clear that big O is taken as $n\to0$, rather than $n\to\infty$ as is more usual for us. For a function $f(x)$ defined on $(0,1)$, we say that $f(x) = O(1/x)$ if there exists a constant $C&gt;0$ such that $f(x) &lt; C/x$ (or, depending on the exact definition, $|f(x)| &lt; C/x$.

This definition is by and large equivalent to the more usual definition which states that $f(x) = O(1/x)$ if there exist two constants $x_0 \in (0,1)$ and $C&gt;0$ such that $f(x) &lt; C/x$ for all $x \leq x_0$.

If you are confused, note that $f(x) = O(1/x)$ under these definitions iff $g(n) = O(n)$ under the usual definition, where $g(n) = f(1/n)$.

For $n$ in this range, we usually use symbols like $\epsilon$ or $\delta$ to emphasize that we are taking the asymptotics as the variable tends to 0 rather than as the variable tends to infinity.