I don't really know how you want it to be proven.. but I'll just do something trivial:
By definition the square root of a positive real number gives 2 real identical numbers (of opposite magnitudes). I.e there exists 2 real numbers x = a, such that x2 = 2 (and x2 = 2, where x = -a). Now 2 is a positive real number, hence it's square roots are real and they exist.
By definition the square root of a positive real number gives 2 real identical numbers (of opposite magnitudes). I.e there exists 2 real numbers x = a, such that x2 = 2 (and x2 = 2, where x = -a). Now 2 is a positive real number, hence it's square roots are real and they exist.