How to show that a "Big-Oh" set is a subset of another?

Question

Let's say we have two "Big-Oh" sets called $\text{Constant}$ and $\text{Logarithmic}$, such that one has $O(1)$, and the other has $O\big(\log(n)\big)$, respectively, how would I show that $\text{Constant} \subseteq \text{Logarithmic}$?

Answer 1

Higher order set contains lower order set. In your case $\lim \limits_{n\rightarrow \infty}\frac{\log n}{1}=\infty.$

Answer 2

Hint:

Logarithms are increasing and unbounded. But a constant is bounded.