Lesson 1

Existence proofs may also be non-constructive: in such arguments, we establish the existence of an object without explicitly exhibiting it. A common approach is proof by contradiction: we assume that the desired object does not exist and derive a contradiction.

For example, it is easy to show that at least one of the numbers \(\sin^{2}(10^{1000})\) and \(\cos^{2}(10^{1000})\) is at least \(1/2\). Indeed, their sum equals one, so at least one of them must be at least \(1/2\) (otherwise, if both were less than \(1/2\), their sum would be less than one, a contradiction). However, determining which of the two satisfies this inequality is difficult (it would require very precise knowledge of the value of \(\pi\)).