How can I find the supremum of $\frac{nt}{1+nt} - \frac{mt}{1+mt}$, where $n$ and $m$ are any natural numbers and ${t}\in{[0,1]}$?
In particular, can I consider this as a maximization problem $max_{{t}\in{[0,1]}}\frac{nt}{1+nt} - \frac{mt}{1+mt}$, meaning that I can take the derivative with respect to $t$, equate it to zero and plug the value of the resulting value of $t$ into the expression that I maximize? I remember that one of my instructors used this kind of technique to find the supremum of a certain set, but I do not know in which sets we can use maximization, and I want to know the logic behind this process if it is valid.