Suppose an inertial observer O' is moving along the positive x axis of an inertial observer O with uniform velocity
At
the origins of both inertial frames coincide. The world line of O' in the reference frame of O is shown below. Each observer carries a clock which are synchronized at the moment the origins coincide.
The clock carried by observer O ticks at times
each tick results in a light signal being sent to O' which is received at times
and instantly reflected back to O.
Because the world line of O' in the inertial frame O is a straight line, there will be a linear relationship between
and
and because it passes through the origin, the relation is of the form 
so that
Similarly, when light signal is received in O' and reflected,
The signals are received at
so that
The signals take equal times to travel from O to O' and back, so the first signal is received by O' at a time in Os frame given
and
Hence the speed of O' relative to O is
so
Hence
and the frequency
of light received by a moving observer is reduced by the factor