Extreme operations can be regarded as infinite arithmetic operations (addition, subtraction, multiplication and division).

If a rational number (fraction) performs infinite arithmetic operations, the result may be irrational (possibly irrational).

In order to have the result of limit operation, the result of limit operation of rational number is called real number (including rational number and irrational number).

When a real number performs limit operation, the result is still in the range of real numbers, which is called the continuity (completeness) of real numbers.

2. There are six equivalence theorems for the continuity of real numbers, including the three you mentioned, which can prove each other.

There is too much content. Look up the mathematical analysis book.

The equivalence between the three theorems and the continuity of real numbers lies in that the operations made by these three theorems can all be classified as arithmetica infinitorum operations (limit operations).

For example, in monotone series, An+ 1 is a little more (less) thAn an. Because of its boundedness, the number of additions (subtractions) is a little less than last time, so the definition of real numbers can guarantee the result.

The same is true of closed interval sets, which are multiplied and subtracted, and neither side is out of bounds.

The method of lower supremum is similar to monotone bounded sequence, and the definition of real number can ensure the lower supremum.

Answer supplement

/sxfx/taolun/5.doc

This is a document about the basic theorem of real number continuity. Go and see if it helps. I haven't learned a few points, and that's all I can help, hehe. When I was studying advanced mathematics, I had a headache with those proof questions! ! !