The observer starts moving towards the light source. The wavecrests start hitting him more frequently - the frequency he measures increases - which means that the speed of the wavecrests relative to the observer increases as well, in violation of Einstein's relativity:

"Doppler effect - when an observer moves towards a stationary source. ...the velocity of the wave relative to the observer is faster than that when it is still."

https://www.youtube.com/watch?v=bg7O4rtlwEE
"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."

http://physics.bu.edu/~redner/211-sp06/class19/class19_doppler.html
"Vo is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + Vo. [...] The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."

http://a-levelphysicstutor.com/wav-doppler.php
There is an ad hoc assumption that saves Einstein's relativity but it is idiotic: When the initially stationary observer starts moving towards the light source with speed v, its motion somehow changes the wavelength of the incoming light - from λ to λ'=λc/(c+v). The idiocy is too great, even for the standards of Einstein's schizophrenic world, so Einsteinians don't discuss it explicitly. Here is an exception:

Professor Martin White, UC Berkeley: "...the sound waves have a fixed wavelength (distance between two crests or two troughs) only if you're not moving relative to the source of the sound. If you are moving away from the source (or equivalently it is receding from you) then each crest will take a little longer to reach you, and so you'll perceive a longer wavelength. Similarly if you're approaching the source, then you'll be meeting each crest a little earlier, and so you'll perceive a shorter wavelength. [...] The same principle applies for light as well as for sound. In detail the amount of shift depends a little differently on the speed, since we have to do the calculation in the context of special relativity. But in general it's just the same: if you're approaching a light source you see shorter wavelengths (a blue-shift), while if you're moving away you see longer wavelengths (a red-shift)."

http://w.astro.berkeley.edu/~mwhite/darkmatter/dopplershift.html
Another exception:

Kip Thorne (4:56): "If you move toward the source, you see the wavelength shortened, but you don't see the speed changed." Kip Thorne - What is Space-Time?

https://www.youtube.com/watch?v=mvdlN4H4T54&t=388s
Pentcho Valev