What's the difference between $O(1)$ and $o(1)$?
A function is $O(1)$ if it is bounded from above. It is $o(1)$ if it tends to zero. Any function which is $o(1)$ is also $O(1)$.

For example, $\sin^2 x$ is $O(1)$ but not $o(1)$, and $1/x$ is both $O(1)$ and $o(1)$.