Suppose that the frame is not inertial, then we must accept that light rays would, in general, travel in curved paths. However, if we observe those light rays from an inertial frame, they necessarily travel in straight lines. Hence, if we observe from this same inertial frame, free objects moving between the same two points as the above light rays it follows that some free objects must travel curved paths. But these objects were assumed free of external influence and thus, as established above, must travel in straight lines in any inertial frame. Hence we have arrived at a contradiction and must conclude that the frame is inertial.
Some useful results are immediate.
An inertial frame is one in which Newton's First Law is valid, i.e. one in which an object free of external influence continues in its state of rest or uniform motion in a straight line. Any inertial frame moves at a constant velocity with respect to any other inertial frame. Observe though that the statement makes no sense as a definition because, until inertial frames have been defined, and a system of measurement prescribed, we can not meaningfully talk about uniform motion in a straight line. Since this object must move with constant velocity as observed in any other inertial frame, the frame itself must also move with constant velocity.