I feel that you don’t mention a logical step between your proof and the point of the “the true meaning of entropy section”. Because you do not make explicit how you use the result that you prove: since S does increase sometimes (and thus is not constant), then by the contraposition of your result, it cannot increase for every state. This becomes clear after rereading the section a couple of times, but I think you can probably make this smoother by explicitly saying it.
Other than that, your explanation makes sense to me, but I’m far from very good at thermodynamics.
I feel that you don’t mention a logical step between your proof and the point of the “the true meaning of entropy section”. Because you do not make explicit how you use the result that you prove: since S does increase sometimes (and thus is not constant), then by the contraposition of your result, it cannot increase for every state. This becomes clear after rereading the section a couple of times, but I think you can probably make this smoother by explicitly saying it.
Other than that, your explanation makes sense to me, but I’m far from very good at thermodynamics.