First of all, the concept of equation

When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation.

Second, the idea of classified discussion

When a problem may lead to different results because of different situations of a certain quantity or number, it is necessary to discuss the various situations of this quantity or number in categories. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A in different categories.

Third, the idea of implicit conditions.

Conditions that are not explicitly stated but can be inferred from existing explicit expressions, or conditions that are not explicitly stated but are routines or truths. For example, in an isosceles triangle, a line segment passing through the vertex is perpendicular to the bottom, so the straight line of this line segment also bisects the bottom and the vertex.

Fourth, analogical thinking

Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.

V. Extreme thoughts

The idea of limit is the basic idea of calculus, and a series of important concepts in mathematical analysis such as continuity, derivative and definite integral of function are defined by means of limit. If you want to ask, "What is the theme of mathematical analysis?" Then it can be summed up as follows: "Mathematical analysis is a subject that studies functions with extreme ideas".