The initially stationary observer starts moving towards the light source, with speed v. Two hypotheses are conceivable:

Hypothesis 1 (Newton's emission theory): The speed of the light relative to the observer shifts from c to c'=c+v. Accordingly, the frequency measured by the observer shifts from f=c/λ to f'=c'/λ.

Hypothesis 2 (Einstein's special relativity): The speed of the light relative to the observer does not shift. Accordingly, the frequency measured by the observer does not shift either.

Needless to say, the experiment (measurement of the Doppler effect) confirms Hypothesis 1 and refutes Hypothesis 2:

http://www.hep.man.ac.uk/u/roger/PHY.../lecture18.pdf
"Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/λ waves pass a fixed point. A moving point adds another vt/λ. So f'=(c+v)/λ."

http://physics.bu.edu/~redner/211-sp...9_doppler.html
"Let's say you, the observer, now move toward the source with velocity vO. You encounter more waves per unit time than you did before. Relative to you, the waves travel at a higher speed: v'=v+vO. The frequency of the waves you detect is higher, and is given by: f'=v'/λ=(v+vO)/λ."

Pentcho Valev