Constructive proof

The idea behind constructive proof is to show the existence of a certain object by describing a method of creating it.

Theorem 2. There exists a rational number x such that \sqrt{10^{100}} < x < \sqrt{10^{100}+1}.

Proof. \sqrt{10^{100}} is 10^{50}. Now, let’s just try different clearly rational values for x that seem to lie between the required boundaries. For example, let’s try x = 10^{50} + 10^{-51}. It’s obviously larger than 10^{50}, so we only need to show that it’s smaller than \sqrt{10^{100}+1}. To do that, let’s compute x^{2} so that we can compare it with (\sqrt{10^{100}+1})^{2}:

x^{2} = (10^{50} + 10^{-51})^{2} = 10^{100} + 2 \times (10^{50} \times 10^{-51}) + 10^{-102} = 10^{100} + 2 \times 10^{50-51} + 10^{-102} = 10^{100} + 2 \times 10^{-1} + 10^{-102}

Clearly:

(2 \times 10^{-1} + 10^{-102}) < 1

therefore

10^{100} + 2 \times 10^{-1} + 10^{-102} < (\sqrt{10^{100}+1})^{2} x^{2} < (\sqrt{10^{100}+1})^{2}

and, by extension

x < \sqrt{10^{100}+1}

Therefore, there indeed exists a rational number x such that \sqrt{10^{100}} < x < \sqrt{10^{100}+1} \blacksquare.