Everybody in the UK has ginger hair.

Proof:

This will be a proof by induction. Let P(n) be the statement: *"n people in the world have ginger hair"*.

Now we show P(n). Clearly we can find one person in the world who is ginger.

Now we assume that if P(n) is true, P(n+1) is true. Get any n+1 people into the UK, get everybody else out. Now, remove one of the people. Now, since P(n) is true (by our assumption), we know that all the n people in the UK are ginger. Now bring the excluded person back in and send a different person out. Again, P(n) is true so all these people are ginger-haired. Bring back the excluded person and we have n+1 people in the UK, all of whom must be ginger.

Therefore, P(n) is true for n arbitrarily large. Specifically, we can make it as large as the number of people in the UK.

So, by induction, all people in the UK are ginger.

]]>** 3 = 1 **

** flaw **

Natural logarithm is only one-one over the reals, over the complex numbers, it is a multi-valued function, so the step is not valid.

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