Final answer:
To prove the condition xy = 1 implies x < 1/2 or y < 1/2, a proof by contradiction is used, leading to the conclusion that at least one of x or y must be less than 1/2.
Step-by-step explanation:
To prove that if x and y are real numbers such that xy = 1, then x < 1/2 or y < 1/2, we can use a proof by contradiction. Assume the opposite of what we want to prove, meaning both x and y are greater than or equal to 1/2. Now, if x and y are both at least 1/2, their product would be x*y ≥ (1/2)*(1/2) = 1/4, which is less than 1. This contradicts our starting condition that xy = 1. Therefore, our assumption that x and y are both greater than or equal to 1/2 must be false. This implies that at least one of them must be less than 1/2.