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?
 A: 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.
A: Young and Freedman (YF) here have a true conclusion but arrived at by an argument that does not work. The conclusion here is that it does not make any sort of physical sense to apply the phrase "observer" or "frame of reference" to something moving at the speed of light. This is correct. However, the argument fails as follows. What YF name "Einstein's second postulate", i.e. the one about the speed of light, is a postulate that applies to observations made with respect to any physically allowed frame of reference. It has nothing to say one way or the other about what would be observed in "a frame of reference moving at the speed of light" (i.e. moving at that speed relative to some allowed inertial frame) because there is no such thing as "a frame of reference moving at the speed of light". You can say the words, but as soon as you try to give meaning to them, by stating how distance and time are to be measured or indicated in such a "frame", you are faced with infinities and impossibilities, such as rigid bodies contracted to zero length, and clocks that do not advance at all. The "impossibility" here is then in giving meaningful definitions of space and time measurements using instruments of no length and no evolution.
It is a good and useful lesson to know that the notion of a frame traveling right at the maximum speed is an unphysical notion, but the way to prove this is not the way suggested by Young and Freedman. Their argument does indeed suffer from the kinds of defects that commenters here have pointed out. 
A: The statement "...Einstein's second postulate also asserts that the headlight beam moves at a speed $c$ relative to the spacecraft,..." is only true if it is qualified by adding the phrase "from the spaceship's point of view." In that case it doesn't contradict what's said about $1\ m/s$ guy's experience. The proof fails. What the postulate actually asserts is in effect that any inertial observer will deem the speed of light to be $c$ regardless of the motion of the source of the light relative to the observer. Bearing that in mind...
If we disregard the requirements for being a proper observer and forget about the fact that we can't observe light that doesn't come directly at us, we can still discuss the matter so as to reconcile apparently contradictory statements by stating them carefully and considering the consequences of their simultaneous application.
We can ask in general whether the relative positioning of two objects moving along the same axis in the same direction at speed $c$ undergoes change. For the sake of simplicity and plausibility, we can suppose that the objects about which we ask are photons. OK then, can one photon catch up to another?
The guy moving somewhere at $1\ m/s$ must say no, that their separation remains the same, that neither catching up nor pulling away occur. But the forward photon must say yes (if we don't consider Polhode's comment so that we can continue to examine other implications) in order to speak in accordance with the postulate invoked above.
However, the object that catches up to the forward photon will be nothing other than a phantom fulfillment of the postulate because the frequency of its cycle of action will be zero (i.e., will be 'Dopplered' to death) so that it will have no energy at all and thereby be undetectable by the forward photon (or by any other entity moving as it does at $c$). This continues to be so as the formerly aft photon  overtakes the other (in theory) and moves away from it to the fore at speed $c$. All this is from the 'cerebration' of the overtaken photon, of course. If that
photon reported only tangible perception, he'd say that he knew nothing of another photon, which would accord with $1\ m/s$ guy's assertion that the aft photon never caught up with the forward one.
If we consider the scenario only from the moment of overtaking and thereafter, we have a model of the original problem as seen by $1\ m/s$ guy and the spaceship, respectively. Reconciliation all around.
