There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ends?
I have been searching for a good explanation on various textbooks at different levels, but none of them provided me a satisfactory explanation. They either
State directly "we assume the string is massless and the pulley has no friction, therefore the tension must be the same throughout the string" without further any elaboration and continue the calculation,
Claim without any reasons that "whenever the rope and the pulley are massless and frictionless, the tension will be the same on both sides", or
Explain using a lot of hand-waving arguments regarding infinite acceleration, non-zero moment of inertia, etc. , in one line or two.
Although those hand-waving arguments of type $(3)$ do sound quite reasonable to me and will convince me for a while every time I review them. But after a while, I will start feeling not well about this whole tension concept again.
Therefore, I am looking for a rigorous (in the physicist's sense) proof of exactly why and when do the tension in rope will be uniform, and hopefully from that I can clean up all of my conceptual difficulties.