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Please help me understand my mistake(s) in this reasoning. I will write as if I knew exactly what I'm talking about for the sake of clarity, but I of course don't.

Consider observer A and its inertial frame, in which it is at rest.

Now, consider a light source B, moving at nearly the speed of light (with respect to the reference frame A), directly towards the observer. Let's call this velocity $v$. Finally, let A and B initially be a distance of one light-year apart, as seen in frame A.

By the time the very first photons (image) emitted by B arrive at A's eyes, B will have travelled the fraction $\frac{v}{c}$ of the distance. Since A is seeing B's first photons, it sees B's image from when it was the entire light year away.

From that point, it will take less than a year for B to reach A, assuming v is big enough. While A will still see the whole of B's trip, accelerated by a Doppler shift, it will also see B coming towards him at a velocity greater than that of light, based on the images he is creating from the photons he receives.

To be extra clear, I am not referring to the velocity of B itself, or the velocity of the photons emited by B, in frame A. I am referring to the velocity at which the observer sees B approach him, based on the timing photons hit his eyes. In other words, I am thinking about the velocity of B's image, which by my understanding can be greater than $c$.

What am I missing?

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Since A is seeing B's first photons, it sees B's image from when it was the entire light year away.

Right.

From that point, it will take less than a year for B to reach A

Sure.

see B coming towards him at a velocity greater than that of light

Yup. They will see the entire trip take a short time, say a few months to cross the light year.

What am I missing?

Nothing at all!

What one person sees vs. another is precisely the whole issue of relativity. You can have very different ideas of the same physical events.

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