← Table of contents →

Select a white background

Backward Proofs

A backward proof uses reverse implications. Instead of thinking of one fact as implying another, you think of one goal as being implied by another or equivalent to another. If you choose to do a backward proof, it is critical that you make it very clear that your proof is backwards. If you do not say anything, a proof is assumed to be forward.

Backwards proofs are closely related to proofs by contradiction, and it is usually easy to convert one of those proofs into the other.

Example

Let's reprove that the arithmetic mean of two different positive numbers is greater than their geometric mean.

Theorem. If x and y are two different positive real numbers, then (x+y)/2 > √xy.

Proof.

(x+y)/2 > √xy ≡ (x+y)2/4 > xy. (since x and y are positive)
≡ (x+y)2 > 4xy. (by algebra)
≡ x2 + 2xy + y2 > 4xy (by algebra)
≡ x2 − 2xy + y2 > 0 (by algebra)
≡ (x − y)2 > 0 (by algebra)
≡ true (since x ≠ y and the square of a nonzero number is positive)

Since our claim is equivalent to true, it must be true.

♦