The process of thermal contact between two objects at different temperatures in an isolated system is an irreversible process, hence the entropy of the system will increase during this process. Existing textbooks do not provide a rigorous proof for this conclusion. Lima's published paper offers an elegant and concise proof, but there are still two issues with the proof: 1. It assumes that the heat capacity of the object is a constant, which is not true in most real situations. 2. The proof process uses an inequality about the concavity of a function's curve, which is still relatively obscure to beginners. This paper provides two alternative proofs that are more rigorous and intuitively simple for the first issue, and for the second issue, it offers another proof using elementary geometric methods for reference by beginners.