Chapter 4.7, Problem 11ES

Let T be the statement: The sum of any two rational numbers is rational. Then T is true but the following “proof” is incorrect. Find the mistake.
“Proof” by contradiction: Suppose not. That is, suppose that the sum of any two rational numbers is not rational. This means that no matter what two rational numbers are chosen their sum is not rational. Now both 1 and 3 are rational because 1=1/1 and $3=3/1,$ and so both are ratios of integers with a nonzero denominator. Hence, by a supposition, the sum of 1 and 3, which is 4, is not rational. But 4 is rational because $4=4/1,$ which is a ratio of integers with a nonzero denominator. Hence 4 is both rational and not rational, which is a contradiction. This contradiction shows that the supposition is false, and hence statement T is true.”