How can one prove from observations of the starry sky that the Earth rotates around its own axis and that the rotation occurs from west to east?

If the Earth did not rotate, then the frame of reference associated with a point on its surface would be inertial (IFR). In IFR, free bodies, which can be assumed to be stars, would remain motionless on the celestial sphere.
The existence of the diurnal rotation of the starry sky unambiguously indicates the rotation of the topocentric reference system, that is, the rotation of the Earth. The daily movement of celestial bodies from east to west indicates the rotation of the earth from west to east.
The existence of the diurnal rotation of the starry sky unambiguously indicates the rotation of the topocentric reference system, that is, the rotation of the Earth. The daily movement of celestial bodies from east to west indicates the rotation of the earth from west to east.

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