Why add a +1 to the constant proving an O(n) bound?
We want to find a $c$ such that $4+4n\le cn$ for all $n$ greater than or equal to some bound $N$. So we need a $c$ such that $(4+4n)/n \le c$, or $(4/n)+4\le c$. Now while it's true that
$$
\lim_{n\rightarrow\infty}\frac{4}{n}+4=4
$$
we can't use $c=4$ since for all positive $n$ we have $(4/n)+4&gt;4$. In other words, we need to pick $c$ to be some number larger than 4 and 5 is as good a choice as any in this case. There's nothing magical about picking a $c$ one greater than the limit; it's just one choice out of infinitely many.