I will use unprimed coordinates to denote the frame in which the light sources A and B are stationary, and primed coordinates to denotes the frame in which A and B have velocity u.
If we start out making the assumption that the source A is at the origin in the unprimed coordinates, then
Without losing generality, we will also assume that the pulses of light are emitted right as the original coordinate system's origin aligns with the origin of the primed system. With this assumption we have
The position of the photon emitted by A in primed coordinates =
The position of the photon emitted by B in primed coordinates = Distance from A to B in primed coordinates +
By the postulates relativity, c is constant for all inertial reference frames. However, we must remember that time will now be different at A and B according to the primed coordinate system.
So we must determine the following three quantities:
3) Distance from A to B as seen from the primed coordinates
The Lorentz Eqautions tell us
1) (remember, we have )
2) (using here)
3) If you use the Lorentz Equations to find and , then compute (which is the distance between A and B in the primed coordinates) we should get
Then the distance between the photons relative to the primed system is
If we do a little algebra to the last line, we should get the desired result.
Does this help? Let me know if anything is unclear.