Is $\sup$ and $\inf$ of a set single?

Question

Lets say we have a set & its $\sup$ is $30$ whether $\max$ exists or no, is it safe to say $30$ is the ONLY $\sup$?

Answer 1

The supremum and infimum of a set is unique. This can be proven.

Take a set $S$, with supremum $\alpha$. Suppose the supremum is not unique, and $S$ has a second supremum, $\beta$. From the definition, then, $\alpha, \beta$ are the least upper bounds. That is, in the set $U(S) := \{ x \mid x \ge s \; \forall s \in S \}$ of upper bounds of $S$, $\alpha \le x$ $\forall x \in U(S)$, and the same is true for $\beta$. However, $\alpha,\beta \in U(S)$ as well (after all, the "least" upper bound is still an upper bound). Thus, in particular, $\alpha \le \beta$ and $\beta \le \alpha$ (they have to be less than or equal to all of the other upper bounds, including each other). However, this implies $\alpha = \beta$, which shows that the supremum is actually unique.

You can perform a similar argument to show uniqueness of the infimum.