Upper Bound vs. Least Upper Bound

Every least upper bound is an upper bound, however the least upper bound is the smallest number that is still an upper bound. Example: Take the set $(0,1)$. It has $2$ as an upper bound but clearly the smallest upper bound that the set can have is the number $1$ and hence it's the least upper bound.

Maybe you like this definition better?

Let $A$ be nonempty. We say that $\alpha$ is the least upper bound of $A$ if

$(1)$ It is an upper bound of $A$, that is, if $x\in A$; then $x\leq \alpha$.

$(2)$ If $\beta$ is any other upper bound $\alpha\leq \beta$. That is, $\alpha$ is the least of all upper bounds of $A$.

As you can see the l.u.b. has the unique property $(2)$. Why unique? Because if $\gamma$ is another l.u.b., by definition, we must have both $\alpha\leq \gamma$ and $\gamma\leq \alpha$, but this means we must have $\alpha=\gamma$. So l.u.b.s when they exist, are unique.