This is Problem 4.20 (g) of the textbook "Reading, Writing, and Proving - A Closer Look at Mathematics", Chapter 4, 2nd ed. Springer, 2003 by Ulrich Daepp and Pamela Gorkin.
Decide whether sentence (3) is true if sentences (1) and (2) are both true. Give reasons for your answers.
Let $M$ and $n$ be real numbers.
(1) If $n > M$, then $n^2 > M^2$.
(2) We know that $n < M$.
(3) So $n^2 \le M^2$.
I am confused by the following three different ideas:
- (3) is false.
Reason: Take $n = -2, M = -1$.
- (3) is true.
Reason: From (1), we can conclude that $n > 0$. Together with (2), we have $0 < n < M$. Therefore, we have $n^2 < M^2$, which implies $n^2 < M^2$.
- We cannot decide whether (3) is true or not.
Reason: (1) can be formulated as $n > M \to n^2 > M^2$, while (2) as simply n < M. From these two formulas, we cannot logically conclude either $n^2 \le M^2$ or $n^2 > M^2$.
Which is correct? What is your opinion?
Specifically, I am not sure whether we are allowed to make reasonings based on our knowledge about real numbers, as done in the second idea. What do you think of it?