(1) Omit (or add) some items;
(2) Enlarge or reduce the numerator or denominator in the fraction;
(3) Scaling by applying basic inequalities.
The theoretical basis of scaling method mainly includes:
Transitivity of 1. inequality;
2. Equal amount plus unequal amount is unequal amount;
3. Comparison of two fractions with the same numerator but different denominator.
Scaling method is a thinking method that guides the direction of deformation through inequality proof.
Generally speaking, the key to scale is "gathering". Of course, it is not random, but purposeful. This means that you must find your scale model. In fact, it is easy to produce an inequality. It is not enough to find an equation and delete something. What you have to do is to find out the original equation as much as possible. If you really love mathematics and are willing to study it, then I suggest you try to expand your mathematics. Especially learn more about some famous equations (if you have time, you might as well refer to some university books, which I did in high school). When you know more about mathematics, you will look back at those strange inequalities, so you are likely to think from a higher angle, which will be very helpful for you to come up with the "identity" really hidden behind that inequality.
Of course, what I said above is the inequality for sequence (this is the most difficult). Before that, you should master some common inequalities and some simple scaling methods. Of course, you should try to master Cauchy inequality and other inequalities, which is good for solving problems. In short, the key is that you should always keep your eyes on the target and "get close" to it. This is the key to the application of scaling method, but unfortunately there is no fixed routine. Therefore, it sometimes takes some "luck" to solve such problems. But with more practice, you will naturally find the feeling.