Should O(1) necessarily stand for a non-zero constant?
$O(1)$ does not represent time. It represents a class of functions, which could be used to measure time, space or literally anything else.

$f(x)\in O(1)$ if, and only if, there is a constant $c$ such that $f(x)&lt;c$ for all large enough&nbsp;$x$. There's no requirement that $f$ must be eventually non-zero. $f(x)\in o(1)$ if, and only if, $\lim_{x\to\infty} f(x) = 0$ so, in particular, for all large enough $x$, $f(x)&lt;1$. Therefore, $f(x)\in O(1)$.