# Understanding tension based on assumptions of pulley system

If we consider a simple pulley system with two masses hanging on each end of a MASSLESS and INEXTENSIBLE string around a MASSLESS and FRICTIONLESS pulley, how then can one reason that the tension at each end of the string must be the same?

My own reasoning:

MASSLESS ROPE means that for any segment of the rope with tension $T_1$ and $T_2$ we have that $\sum F = T_ 2 - T_1 = 0$ (since $m = 0$) and thus the tensions must be the same, on a non curved rope at least!

INEXTENSIBLE means that no energy can be stored in the string, however I fail to see how this is a neccesary condition (for equal tension)

MASSLESS PULLEY means that no rotational inertia exists, and thus no force can alter the tension of the string (?)

FRICTIONLESS PULLEY is hard for me to figure.

Needless to say, I feel quite at a loss conceptually!