dehammer, maybe you understand what you are saying, but you say it in such a way that it is confusing. Just agree on the following:

1. both observers will measure the same light speed c relative to their reference frames respectively; even though they move relative to each other.

2. Each observer will see a Doppler shift when looking at the other observer and the light moving from that observer, but this does not mean that any of the observers are actually catching up with light within their respective reference frames.

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