Final answer:
By proving the contrapositive, we have shown that if the product of x and y is at least 2, then at least one of the numbers must be greater than or equal to 1.
Step-by-step explanation:
To show that if xy ≥ 2 (where x and y are real numbers), then x ≥ 1 or y ≥ 1, we can use a proof by contraposition. The contrapositive of the given statement is: If x < 1 and y < 1, then xy < 2. We will prove this contrapositive.
Assume x < 1 and y < 1. Since both x and y are less than 1, their product xy will be less than x and also less than y (as multiplying two positive numbers less than 1 will result in a number smaller than either factor). Therefore, xy will be less than 1*1=1, which is certainly less than 2.
Since we have shown the contrapositive to be true, by valid deductive inferences such as modus tollens, the original statement must also be true. Hence, it is indeed the case that if xy ≥ 2, then x ≥ 1 or y ≥ 1.