Before asking this question, I tried checked many sources on the internet but couldn't find an answer for my question.
What does it mean that a notation has a total order ?
For example, if we have f(n) = ϴ g(n), that includes g(n) = ϴ f(n) which means that the "ϴ" notation has a total order.
But to prove the opposite for "Big-Oh" notation, I can't think about 2 functions where f(n) is not equal to O g(n) and at the same time g(n) is not equal to O f(n).
Because if we calculate the limits of 2 functions and the result is an R positive, that includes that f(n) = O g(n) AND g(n) = O f(n), doesn't that already prove that big Oh is total order ? ( Works in both directions )
To have two incomparable functions, you need $f$ to be larger than $g$ by more and more sometimes and $g$ to be larger than $f$ by more and more sometimes. Let me define two functions from the naturals to the naturals. $$f(n)=\begin {cases} 1&n\text{ even}\\n&n\text{ odd} \end {cases}\\ g(n)=\begin {cases} n&n\text{ even}\\1&n\text{ odd} \end {cases}$$ You can verify that neither one is bounded by a constant times the other. This demonstrates that big O is not a total order.