Fix any errors in my proof (beginner)
Fix any errors in correctness, structure, etc.
Theorem. Suppose that a and b are nonzero real numbers. Prove that if $a < 1/a < b < 1/b$, then $a < −1$.
Proof. By contrapositive, if $a \geq1$, then it should not be the case that $a < 1/a < b < 1/b$. Consider when a is 1 and b is $\frac{1}{2} $. The statement then would be $1 < \frac{1}{1} < \frac{1}{2} < 2 $. In this case, $\neg (a < \frac{1}{a} < b < \frac{1}{b}) $ because $1 = \frac{1}{1} $. Therefore, because the contrapositive is true, it must be the case that if $a < 1/a < b < 1/b$, then $a < −1$.
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