Can the supremum of a set be infinity?

Let T be a totally ordered set. The supremum of a subset S ⊆ T , if it exists, is an element s ∈ T such that (1) s is an upper bound of S , and (2) for all x which are upper bounds of S , x ≥ s . You can prove that if such an s exists then it is unique using the properties of a total order. read more

The idea of supremum and maximum come only for a bounded set. You are considering the set $\{n : n \in \mathbb{N}\} = \{1, 2, 3, \dots\}$. This is an unbounded set in $\mathbb{R}$, as for any positive real number $G$, you shall get an element $m \in \mathbb{N}$ s.t. $m > G$. read more

But the supremum does not have to be in a set in which case the set would not have a maximum. The supremum and infimum of the intervals (0,1), [0,1), (0,1], and [0,1] are 0 and 1 respectively for all four intervals. The maximum (largest number in the set) of (0,1] and [0,1] is 1 but (0,1), [0,1) do not have a maximum. read more