The empty said is bounded because you can find a value for which there are no elements in the set greater than said value, and a value for which there are no elements in the set smaller than said value. However, if you define "bounded above" as requiring the set to have an element that is greater than all other elements in the set (which seems to be what you did in your original post), then the result is false because no such element can exist. But the usual definition does not require the bounds to be within the set itself. For example,
has an upper bound (and supremum) of 1, but there is no element within the set
that is greater than all other elements within the set. This is why you need to be careful with your definitions.