Forward reasoning is starting with the premises, and using the premises together with axioms and known theorems to take a sequence of logically connected steps to lead to a conclusion (for a direct proof). Similarly, with indirect reasoning you can start with the negation of the conclusion and obtain the negation of the premises.
Backward reasoning to prove a statement is to find a statement that we can use to prove .
Given two positive real numbers and , their arithmetic mean is and their geometric mean is . When comparing the arithmetic and geometric means of pairs of distinct positive real numbers, we find that the arithmetic mean is always greater than the geometric mean. Can we prove this inequality is always true?
To prove that when and are distinct positive real numbers, we can work backward. We construct a sequence of equivalent inequalities. The equivalent inequalities are
Because when , it follows that the final inequality is true. Because all the inequalities are equivalent, it follows that when . Once we have carried out this backward reasoning, we can build a proof based on reversing the steps (into forward reasoning) (Note that the steps of our backward reasoning will
not be part of the final proof. These steps serve as our guide for putting this proof together.)
Proof: Suppose that x and y are distinct positive real numbers. Then because the square of a nonzero real number is positive (see Appendix 1). Because , this implies that . Adding to both sides, we obtain . Because , this means that . Dividing both sides of this equation by , we see that . Finally, taking square roots of both sides (which preserves the inequality because both sides are positive) yields . We conclude that if x and y are distinct positive real numbers, then their arithmetic mean is greater than their geometric mean .
We've already done that a few times elsewhere in this wiki (where?). For example, using techniques of proving is irrational to prove is irrational, or the cube root of .