Reduction to absurdity is a common method in mathematics, and some propositions can only be proved by it. Here is a brief introduction. The following steps are often used to prove a proposition by reduction to absurdity: 1) Assuming that the conclusion of the proposition is not valid, 2) Reasoning, in which one of the following situations occurs: it contradicts the known conditions; Contradictions with axioms or theorems, 3) Due to the appearance of the above contradictions, it can be asserted that the original assumption that "the conclusion is not valid" is wrong. 4) Affirm that the conclusion of the original proposition is correct. Proving a proposition by reduction to absurdity is actually such a thinking process: we assume that "the conclusion is not established", and once the conclusion is not established, problems will arise, which is caused by contradictions with known conditions; Exposed in a way that violates axioms or theorems. How did this problem come about? There is no mistake in reasoning, and there is no mistake in known conditions, axioms or theorems, so the only mistake is the original assumption. "If the conclusion is not established, there must be a correct conclusion. Since there is something wrong with "the conclusion is not valid", then the conclusion must be valid. Reduction to absurdity is also called reduction to absurdity. The British mathematician Hardy (1877- 1947) gave an interesting comment on this proof. In the chess game, a strategy is often adopted, which is called "abandoning one's son to gain the upper hand", that is, sacrificing some pieces in exchange for advantages. Hardy pointed out that reducing to absurdity is a strategy far superior to any chess skill. A chess player can sacrifice several pieces, and a mathematician can sacrifice a chess game. Reduction to absurdity is the greatest strategy imaginable. Let's prove the reciprocity of Theorem 1 and Theorem 4. Two propositions need to be proved: (1) Theorem 4 is established by the theorem 1; (2) Establish theorem1from theorem 4; Proof (1). By reducing to absurdity. Let's start with the conclusion of negative theorem 4. Suppose there is, according to theorem 1, there should be, which contradicts the condition of theorem 4. I found the contradiction I wanted. The correctness of the theorem is proved. The reader himself proves that theorem 1 is established from theorem 4. Let's make some explanations from the point of view of set. Let {a continuous function on a closed interval}; = {Function to get the maximum value on a closed interval}. These are two different sets of things. The above theorem tells us that it is a subset of Figure 2. If a function is not in the middle, it must not be in the middle. This is the inverse negation theorem. It is the same as the positive definite theorem. Similarly, the inverse theorem is true or false. Thinking questions prove the inverse theorem is true and false. It is very important to understand the structure of the theorem and the four forms of the theorem, which prepares for the following necessary and sufficient conditions research. But this is only one aspect of the problem. To learn the theorem well, we need to consider the following five questions: how to prove the theorem, how to popularize the theorem, how to use the theorem and how to understand the theorem. Hope to adopt, thank you.