There are a couple really cute proofs that I’ve learned along the
way. This one asks if there exist two irrational numbers, and
such that
is a real number.
The interesting thing about this proof is that it only shows existence but does not actually construct it. There are several of these kind of proofs and some people are actually dissatisfied with them. (Some people are against proof by contradiction!) Another famous proof that shows existence without construction is Brouwer’s Fixed Point Theorem.
The proof is actually pretty simple. Set . Now,
consider
. If it is rational, we are done. If not, let
and
. Both
and
are
irrational, but
which is rational.
Neat, isn’t it?
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