untheorem:
Given a room with any (finite) number of people in it, if 1 of them is a redhead, then all of them are.

This is patently false, of course. Nonetheless we have the following "proof" by induction.

unproof:
We'll denote the number of people in the room by n.

• base case: n=1.
This case is clear: it says that for a room with one person in it, if one of them (and there's only one) is a redhead, then all of them are redheads.
• induction case: we shall suppose the untheorem true for n, and prove it for n+1.

Consider a room with n+1 people in it, one of them a redhead. Ask one of the non-redheads to step outside the room for a moment. Then we are left with a room with n people in it, and one of them is (still) a redhead. By the induction hypothesis, all n people are redheads!

Now ask the (only) non-redhead who stepped out to return to the room; ask some other person to step out, and again 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. Returning the redhead who stepped outside to the room, we see that all n+1 people are redheads.

By the principle of mathematical induction, the theorem is proved (or rather, the untheorem is unproven).

What's wrong here??