The following writeup is erroneous. I leave it here purely for the benefit of other fools like me who do not understand induction (and as of now, nine people have upvoted this, so there are some). Details at the end.

The mistake that glared at me when I read the original writeup came here:

"...we're left with n people in a room, one of them (in fact, all but one of them!) a redhead. By the hypothesis, we deduce that all n are redheads, in particular the remaining person."

(i.e. 1 person in room is a redhead, hypothesis states that if 1 person is redheaded all people in the room are, therefore all people in the room are redheaded)

This seems to be using the original "untheorem" to prove itself; you can't do that. This is a logical fallacy known as "begging the question". If it was valid, you could make the following, simpler argument:

Hypothesis: If one person in the room is redheaded, all of them are
Presume the existence of a room with n people in it, one of them redheaded
By the hypothesis, if one person is redheaded, all n of them are
Therefore, if one person is redheaded, all of them are, and the hypothesis is true

(actually, this is essentially the argument form used, as far as I can see, and the rest is simple handwaving)

Swap has informed me, albeit more politely, that I am in fact talking utter bollocks, due to my lack of sound mathematical training. When using induction, I now understand, once you have (un)proved the (un)theorem for the smallest possible value (in this case, one person), you can then take the theorem with regard to n to be true for the purpose of proving the theorem with regard to n+1. Since this was the manner in which the assumption was used, my criticism is wrong.