Common false proofs
The fallacious proofs are stated first and then links to the explanations of their fallacies follow.
2 = 1
Then we have
All horses are the same color
We shall prove that all horses are the same color by induction on the number of horses.
First we shall show our base case, that all horses in a group of 1 horse have the same color to be true. Of course, there's only 1 horse in the group so certainly our base case holds.
Now assume that all the horses in any group of horses are the same color. This is our inductive assumption.
Using our inductive assumption, we will now show that all horses in a group of horses have the same color. Number the horses 1 through . Horses 1 through must be the same color as must horses through . It follows that all of the horses are the same color.
All numbers are equal
Consider arbitrary reals and , and let = . Then