Final answer:
Given a real number x, we can prove that there exist unique numbers n and ɛ such that x = n + ɛ, where n is an integer and 0 ≤ ɛ < 1. We can use the concept of floor and fractional part to demonstrate this. The numbers n and ɛ are unique.
Step-by-step explanation:
In order to prove that given a real number x there exist unique numbers n and ɛ such that x = n + ɛ, where n is an integer and 0 ≤ ɛ < 1, we can use the concept of floor and fractional part. Let n = floor(x) and ɛ = x - floor(x). The floor function returns the greatest integer less than or equal to x, and the fractional part represents the decimal value of x. Using these definitions, we can show that x = n + ɛ.
To prove uniqueness, we can consider the contrapositive. If there exist two different pairs (n1, ɛ1) and (n2, ɛ2) such that n1 + ɛ1 = n2 + ɛ2 = x, then we can show that n1 = n2 and ɛ1 = ɛ2. Therefore, the numbers n and ɛ are unique.