Is A Supremum Unique

The concept of a supremum, or least upper bound, is fundamental in real analysis and related fields. It helps us precisely define the “smallest” value that is still greater than or equal to all elements in a set. But a natural question arises: Is A Supremum Unique? The answer, as we will explore, is a resounding yes, provided it exists.

The Definitive Answer Is A Supremum Unique?

Yes, the supremum of a set, if it exists, is unique. To understand why, let’s consider what a supremum actually is. A supremum of a set S (denoted sup(S)) is an upper bound of S that is also the *least* upper bound. This means two things must be true: first, every element in S is less than or equal to sup(S), and second, any number smaller than sup(S) cannot be an upper bound of S. This “least” property is what guarantees uniqueness. We can demonstrate this by contradiction.

Suppose we have a set S with two different suprema, say sup1(S) and sup2(S), and assume that sup1(S) < sup2(S). Since sup2(S) is the *least* upper bound, it means that sup1(S), which is smaller, cannot be an upper bound for S. But this contradicts our initial assumption that sup1(S) *is* a supremum (and therefore an upper bound) of S. Alternatively, if we assume sup2(S) < sup1(S), the same contradiction arises. This reasoning leads us to a crucial point.

Here’s a summary of key points in understanding that “Is A Supremum Unique”

• A supremum must be an upper bound.
• A supremum must be the *least* upper bound.
• If multiple upper bounds exist, only one can be the smallest.

The above points can be summarised in the following table:

Want to dive deeper into the formal proof and explore related concepts? Look for resources on real analysis and order theory. Textbooks dedicated to these subjects provide rigorous explanations and examples that will solidify your understanding of suprema and their unique properties.