Assuming you are talking about the rationals Q, your set isn't even defined in terms of elements of Q. You should phrase it as x^{2} < 2. Why should it have a least upper bound? There is no theorem stating that a subset of the rationals Q which is bounded above has a least upper bound in Q. In fact, one way to develop the real numbers is to extend them by Dedekind cuts which, effectively, adds all such upper bounds and gives the reals R. Such subsets viewed as subsets of R do have least upper bounds in R.
