I didn't find a demonstration on the web of how to obtain the exponential probability distribution  without using a Poisson distribution, and I don't manage to do it myself.

I am trying to understand how to get this pdf with the only assumption that for each interval dt, the probability of an event is constant. I guess the demo is pretty immediate, but I am missing it.

If the maximum exists, then the supremum and maximum are the same.  However sometimes the maximum does not exist, and there is no maximal element.  In this case it still makes sense to talk about a least upper bound.

The classic example is the set of all rationals whose square is less than or equal to $2$.  That is the set $$A=\left\{ r\in\mathbb{Q}:\ r^{2}\leq2\right\}.$$

$A$ has no maximal element, however it does have a supremum and $\sup A=\sqrt{2}$.

An even simpler example is the set of all reals that are strictly less than $2$:  $$B=\left\{ r\in\mathbb{R}:\ r<2\right\}.$$  This set has no maximum since for any $x\in B$ the element $\frac{x+2}{2}$ satisfies $x<\frac{x+2}{2}<2$.  However it is not hard to see that $\sup B=2$.
