I am given the following definition of the right endpoint $x_F$ of the distribution function $F$:
$x_F = sup\{x \in \mathbb R: F(x) < 1\}$
I know that the right endpoint of a discrete uniform distribution $\mathcal U[1,5]$ is $x_F = 5$. However, this does not make sense to me with respect to the definition above.
The set of which we take the supremum in this example is {1,2,3,4}. As there exists a maximum which is part of the set, the supremum and maximum of the set are the same and $= 4$.
What do I miss here?