Note that $\sim$ as defined in the entry Landau notation is an equivalence relation on the set of all functions from $\mathbb{R}^+$ to $\mathbb{R}^+$ This set of functions will be denoted in this entry as $F$

Reflexive: For any $f \in F$ $\displaystyle \lim_{x \to \infty} \frac{f(x)}{f(x)}=1$ and $f \sim f$

Symmetric: If $f,g \in F$ with $f \sim g$ then $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}=1$ Thus:

Therefore, $g \sim f$

Transitive: If $f,g,h \in F$ with $f \sim g$ and $g \sim h$ then $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}=1$ and $\displaystyle \lim_{x \to \infty} \frac{g(x)}{h(x)}=1$ Thus:

Therefore, $f \sim h$