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1. Homework Statement

1. Homework Statement

Exercise 3.2.8. Suppose that a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1.

## Homework Equations

My proof uses three other facts, the first one of which is proven in a previous exercise. The second and third ones I didn't bother proving since I consider them to be rather trivial.

- ## 0 < a < b \to \frac{1}{b} < \frac{1}{a} ##
- ## a < 0 \to \frac{1}{a} < 0 ##
- ## a < 0 \to -a > 0 ##

## The Attempt at a Solution

"By the contrapositive of ## 0 < a < b \to \frac{1}{b} < \frac{1}{a} ## we can conclude that ## \frac{1}{a} \le \frac{1}{b} \to a \le 0 \vee b \le a ##. But then since ## b > a ## and ## a \ne 0 ##, it follows that ## a < 0 ##. Now suppose that ## -1 < a < 0 ##. Since ## a < 0 \to \frac{1}{a} < 0 ##, we know that ## \frac{1}{a} < 0 ##. Therefore, ## -\frac{1}{a} > 0 ##. This means that we can multiply both sides of the ## -1 < a ## inequality by ## -\frac{1}{a} ## to obtain ## \frac{1}{a} < -1 ##. However, we assumed that ## -1 < a ##. This would mean that ## \frac{1}{a} < a ##, which is a contradiction. Therefore, it must be the case that ## a \le -1 ##. We can also discard the ## a = -1 ## case because we would arrive at ## a = \frac{1}{a} ##, which is another contradiction. Thus, ## a < -1 ##."