Problem: (Stanford Math 51H) Prove that every positive real number has a positive square root (That is, for any , prove that there is a real number such that ). [Usual properties of the integers are assumed.]
Solution: Consider the set . We can show that is non-empty by selecting an integer large enough such that . Since is a real and the integers are unbounded, there exists a positive integer such that , thus so is bounded from above.
Now there must exist such that . We claim that . Suppose the contrary; then either or .
CASE 1: If , then consider . Choose large enough such that
which is definitely true for any . But then and , contradicting the fact that is the supremum.
CASE 2: If , then consider . Again, choose large enough such that
which we know is true for (furthermore, we impose the restriction so our resulting real is positive). Then for all and , contradicting the fact that is the supremum.
Hence we know that , or that the square root of any positive real number exists and is a positive real number. QED.
Comment: This is from the real analysis portion of a freshman honors calculus course, i.e. a rigorous treatment of the real numbers which is the basis for calculus like limits and stuff. Really understanding calculus involves really understanding how the real number system works.
Practice Problem: (Stanford Math 51H) If with , prove:
(a) There is a rational .
(b) There is an irrational .
(c) contains infinitely many rationals and infinitely many irrationals.