methods to tackle this, such as Fermi's method, applying Green's theorem, or even consulting AI tools like ChatGPT. However, what I truly want to see is your proof. To excel, your method should be: $1.2.3.$ Well$-$Defined: Clearly outline the steps and principles used in your derivation. Well$-$Constructed: The reasoning should flow logically and convincingly. Beautiful: The proof should exhibit creativity or elegance in its approach. If you can impress me with an unconventional and ingenious proof, you will earn extra points. A correct proof, even if not particularly beautiful, will still be rewarded, but it won$$t qualify for extra credit. Hints: $$ Hint $1$: Use the Carnot Principles and the Kelvin$-$Planck Statement for the derivation. $$ Hint $2$: Apply rigorous logical reasoning to validate the inequali