Lesser-known fake proofs

Question

I believe I have seen almost every elementary bogus proof of $0=1$ or $1=2.$ I am wondering if there are any lesser-known "proofs" where the error is harder to spot.

Answer 1

There's the classic example of the proof that all horses are the same color.

We prove by induction on $ n $ that for any set of $ n $ horses, all horses in that set have the same color.

Base case: If $ n = 1$ then certainly that horse has the same color as itself.

Inductive step: Suppose that for all sets of $ n $ horses that the result holds. Consider a set of $ n + 1 $ horses, say $ A = \{h_1, \ldots, h_{n+1}\} $. Then every horse in the set $ \{ h_1, \ldots, h_n \} $ has the same color by the inductive hypothesis. Thus, $ h_1 $ has the same color as all horses in $ \{h_2, \ldots, h_n \} $, as does $ h_{n+1} $. Thus, all the horses in $ A $ are the same color.

EDIT: To show that $ h_{n+1} $ has the same color as all horses in $ \{h_2, \ldots, h_n \} $, use the inductive hypothesis on $ A - \{h_1\} $.

Hence, by induction, all horses have the same color, as there are only finitely many horses.

Answer 2

Let $$x=1+1+1+1+.....$$ Then $$x=1+(1+1+1+.....)=1+x$$

Now subtract $x$ from bout sides of the $$x = 1+x$$ to get $0=1$ and add $1$ to this new equality to get $1=2$

Answer 3

What's the sum of $1-1+1-1+\cdots$?

Let $x=1-1+1-1+\cdots.$

1.

$$x=1-1+1-1+\cdots=1-(1+1-1+\cdots)=1-x.$$Thus, $x=\dfrac{1}{2}.$

2.

$$x=1-1+1-1+\cdots=(1-1)+(1-1)+\cdots=0+0+\cdots=0.$$Thus, $x=0.$

3.

$$x=1-1+1-1+\cdots=1+(-1+1)+(-1+1)+\cdots=1+0+0+\cdots=1.$$Thus, $x=1.$

Therefore,$$\dfrac{1}{2}=0=1?$$