Let A and B be non-empty bounded subsets of R. DefineA − B := {a − b : a ∈ A, b ∈ B} .Prove thatinf(A − B) = inf A − sup B

Step 1: Define the variables

Let's define A and B as non-empty bounded subsets of R. We also define A - B as the set of all differences a - b, where a is an element of A and b is an element of B.

Step 2: Prove the inequality inf(A - B) >= inf A - sup B

Let's take any element a from A and any element b from B. Since a is greater than or equal to inf A and b is less than or equal to sup B, we have a - b >= inf A - sup B. This means that every element of A - B is greater than or equal to inf A - sup B. Therefore, the infimum of A - B is also greater than or equal to inf A - sup B.

Step 3: Prove the inequality inf(A - B) <= inf A - sup B

Let's take any ε > 0. Since inf A is the greatest lower bound of A, there exists an element a in A such that a < inf A + ε/2. Similarly, since sup B is the least upper bound of B, there exists an element b in B such that b > sup B - ε/2. Therefore, a - b < inf A - sup B + ε. Since ε is arbitrary, we have inf(A - B) <= inf A - sup B.

Step 4: Conclusion

From steps 2 and 3, we have inf(A - B) >= inf A - sup B and inf(A - B) <= inf A - sup B. Therefore, we can conclude that inf(A - B) = inf A - sup B. This completes the proof.