Question

For every bounded, non-empty set A⊆R, sup{∣a−b∣:a,b∈A}=sup{a−b:a,b∈A}=sup(A)−inf(A)

(Note 1: There are two equalities and they should be proved separately.)
(Note 2: Below are two examples that may help you understand what needs to be proved in the first equality)
Say A={7,10}. Then we have {a−b:a,b\in A}={7−7,7−10,10−7,10−10}={0,−3,3} (note that we have to let a and b range over all possible values in the set A). So sup {a−b:a,b\in A}=3. On the other hand, {∣a−b∣:a,b\ in A}={∣7−7∣,∣7−10∣,∣10−7∣,∣10−10∣}={0,3} and sup {∣a−b∣:a,b\ in A}=3.
A more complicated example is the following: let A=(0,1). Then {a−b:a,b\in A}=(−1,1) and {∣a−b∣ :a,b\ in A}=[0,1) and the supremum of both sets is 1 .

Answer 1

The proof involves showing equivalence between the supremum of absolute differences and the supremum of regular differences within a set A, and additionally proving that the largest difference is the same as the difference between the supremum and infimum of the set A.

Step-by-step explanation:

The student's question pertains to real analysis, specifically the properties of supremum (least upper bound) for differences within a bounded, non-empty subset of real numbers (R). We seek to prove two equalities that reflect the relationship between the set of absolute differences and the set of differences within a subset, and how these relate to the supremum and infimum of the set.

The first equality to prove is that sup{\u007C a-b \u007C : a, b \u2208 A} = sup{a-b : a, b \u2208 A}. The intuition behind this is that for any positive difference a-b (where a > b), there exists a corresponding b-a which is negative and their absolute values will be the same; thus, the supremum in terms of absolute value will be equal to the supremum of the regular differences, as negative numbers won't affect the supremum. The supremum among the absolute differences captures the largest distance between any two points in set A.

The second equality is sup{a-b : a, b \u2208 A} = sup(A) - inf(A). This indicates that the largest difference between any two elements in A is exactly the difference between the supremum and infimum of A itself. This is because the supremum reflects the closest non-exceeding upper bound and the infimum is the greatest lower bound of the set, thereby together providing the maximum span or width of the set.

To prove these equalities, we apply properties of arithmetic operations like addition and subtraction for both positive and negative numbers, as well as consider the properties of uniform distribution and the concept of complements and vector subtraction in broader mathematical contexts.