In a process carried out on a system, the initial pressure and...

For a given thermodynamic system, the state is uniquely determined by its pressure and volume. Since the initial pressure and volume are identical to the final pressure and volume, the system returns to its initial state.

Internal energy and temperature are state functions, meaning they depend only on the state of the system and not on the path taken. Therefore, the initial internal energy must equal the final internal energy, and the initial temperature must equal the final temperature.

Work done and heat exchanged are path functions. In a cyclic process (where the initial and final states are the same), the net work done is the area enclosed by the cycle on a $P-V$ diagram, which is not necessarily zero. By the first law of thermodynamics ($\Delta U = Q - W$), since $\Delta U = 0$, the net heat given to the system equals the net work done, which also need not be zero.