Originally Posted by

**HallsofIvy** Obviously a homework problem.

The only real difficulty is in showing that the set does NOT have a largest member.

Suppose x is the largest member of the set of all rational numbers whose square is less than 2. Since $\displaystyle 1.4^2= 1.96< 2$ you can assume 1.4< x. Since $\displaystyle 1.5^2= 2.25> 2$ you can assume x< 1.5. Let $\displaystyle d= 2- x^2$ (d is, of course, rational). Then $\displaystyle d< 2- 1.4^2= 2- 1.96= 0.04$. Now find an integer, n such that $\displaystyle (x+ d/n)^2= x^2+ 2 dx/n+ d^2/n^2< 2$ showing that x is NOT the largest number in that set.