I am trying to justify the big-O order of a runtime complexity by finding a $c$ and $n_0$ that hold for it. Does the left side of the justification need to be one or higher, or can it be any value so long as it is a real number and less than the right side? For instance, is the following valid:$$n^2/4 + 1/2 \leq c\cdot n^2\,,$$ when $c= 1$ and $n=1$:$$3/4  \leq 1\,.$$
Big-O Justification Question
asymptotics;proof techniques;landau notation
The Big-O notation describes the limiting behavior of a function when  the argument tends towards a particular value or infinity, usually in  terms of simpler functions.-WikipediaIf you see the formal definition, then the only restriction is that the domain of functions must be $\mathbb{R}$. Hence, by definition the range of the functions can at least be any vector space.In the most common use in CS, the range of the functions is also taken to be $\mathbb{R}^+$. To answer your question, by definition, the left side can be any real positive value.