What does "bounded above" mean in Family of Bachmann–Landau notations?
A quantity $a$ is bounded above by a quantity $b$ if $a \leq b$.

A quantity $a$ is bounded above (up to constant factor) by a quantity $b$ if there exists a constant $C&gt;0$ such that $a \leq Cb$. (This makes sense when $a,b$ depend on some other variable).

A quantity $a(n)$ is bounded above (up to constant factor) asymptotically by a quantity $b(n)$ if there exist constants $N,C&gt;0$ such that $a(n) \leq Cb(n)$ for all $n \geq N$. This is usually expressed as $a(n) = O(b(n))$.