http://www.hep.man.ac.uk/u/roger/PHYS10302/lecture18.pdf Roger Barlow, Professor of Particle Physics: "The Doppler effect - changes in frequencies when sources or observers are in motion - is familiar to anyone who has stood at the roadside and watched (and listened) to the cars go by. It applies to all types of wave, not just sound. (...) Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/lambda waves pass a fixed point. A moving point adds another vt/lambda. So f'=(c+v)/lambda."
If "in time t, ct/lambda waves pass a fixed point", and if "a moving point adds another vt/lambda", then the speed of the light waves relative to the moving observer is:
c' = (lambda)(ct/lambda + vt/lambda)/t = c + v
in violation of special relativity. If v is small enough, the relativistic corrections are negligible and both c'=c+v and f'=c'/lambda become exact formulas.