special relativity - Young and Freedman Proof: No Observer Can Travel at the Speed of Light

I have been reading the section on Relativity in the ninth edition of University Physics by Young and Freedman. They include the following proof that no observer can move at the speed of light.

University Physics states:

Einstein's second postulate immediately implies the following result: It is impossible for an inertial observer to travel at c, the speed of light in vacuum. We can prove this by showing that travel at c implies a logical contradiction. Suppose that [a] spacecraft S' [...] is moving at the speed of light relative to an observer [E] on Earth, so that [the velocity of S' with respect to E equals c]. If the spacecraft now turns on a headlight, the second postulate now asserts that the Earth observer E measures the headlight beam to be also moving at c. Thus this observer measures that the headlight beam and the spacecraft move together and are always at the same point in space. But Einstein's second postulate also asserts that the headlight beam move at a speed c relative to the spacecraft, so they cannot be at the same point in space. This contradictory result can be avoided only if it is impossible for an inertial observer, such as a passenger on the spacecraft, to move at c.

The main point of the proof is that the postulate implies an inconsistency in the relative location of objects in space. The Earth observer sees the light beam in the same location as the spacecraft, but the observer in the spacecraft sees the location of the light beam to be ahead of the spacecraft.

But couldn't a similar line of reasoning be used to show that no observer can move at the speed 1 m/s if the postulate is to hold? In this case, the Earth observer sees the light beam ahead of the spacecraft, but the observer in the spacecraft would see the light beam in a location further ahead of the spacecraft. There is an inconsistency in the relative locations of the light beam and spacecraft.

I found that the 1981 book Discovering Relativity for Yourself by Sam Lilly gives a proof with similar reasoning to show an observer cannot move at the speed of light.

What am I failing to understand in this line of reasoning? Why does it apply to observers moving at the speed of light but not other speeds?

Answer

In the case of $1 m/s$, the earth observer would still see the spacecraft and beam of light having a certain growing distance between them, which is coherent with the argument that the observer in the spacecraft measures the light beam having any positive velocity ($c$) relative to him. But in the case of the spacecraft having a velocity of $c$, the earth observer isn't observing any growing distance between them, but the observer in the spacecraft must see the distance growing, because the light beam moves with velocity $c$ relative to him.

Basically the difference is, that if the spacecraft is moving even at $0.99999*c$ the earth observer and the observer in the spacecraft, both see the distance growing.