Question

How was the second law of thermodynamics proved?

Answer 1

The basic laws are some basic postulates of the subject. There is no way you can prove them fundamentally.

Explanation:

Consider, if you want to prove the law of conservation of momentum, you shall need Newton's second and third law.

But, if someone asks you to prove the second or the third law from an even more fundamental concept, you are completely helpless.

The same is applicable to this case. No way to prove the first or second laws of thermodynamics. They make sense and give us experimentally verifiable correct results and that's all we need to know to be sure about their correctness.