Here's a constructive proof that does not rely on the law of excluded middle: https://en.m.wikipedia.org/wiki/Square_root_of_2#Constructive_proof https://en.m.w

Here's a constructive proof that does not rely on the law of excluded middle: https://en.m.wikipedia.org/wiki/Square_root_of_2#Constructive_proof https://en.m.wikipedia.org/wiki/Square_root_of_2#Constructiv...