1. The research objects are different: the compactness theorem mainly studies the properties of real number sequences, while the convergence theorem mainly studies the properties of function sequences. Specifically, the column compactness theorem focuses on whether the real number sequence has some "compactness" characteristics, that is, whether there is a way that the sequence can be covered by a finite number of points; Convergence theorem focuses on whether the value of function sequence near a certain point tends to a certain limit value.
2. Different descriptions of properties: the column compactness theorem describes a global property, that is, for a given real number sequence, we can judge whether it has column compactness; The convergence theorem describes a local property, that is, for a given function sequence, we can only judge whether it converges near a particular point.
3. Different scope of application: Column compactness theorem has a wide range of applications in studying the properties of real number sequences. For example, when proving some important mathematical theorems (such as the Polchano-Weisstras theorem), we need to use the compactness theorem to simplify the problem; Convergence theorem is more widely used to study the properties of function sequences. For example, when learning mathematical concepts such as series and integrals, we need to use convergence theorem to judge whether these concepts are valid.
4. Different proof methods: Because the objects and properties described by the column compactness theorem and the convergence theorem are different, their proof methods are also different. The proof of column compactness theorem usually needs to construct a finite cover by using the completeness or countability of real numbers, thus proving that a given real number sequence has column compactness; The proof of convergence theorem needs to use the definition of limit or Cauchy criterion to prove that a given function sequence converges to a certain point.
In a word, although compactness theorem and convergence theorem are important concepts in mathematical analysis, they are obviously different in research object, property description, application scope and proof method. Understanding these differences will help us better understand and master these two concepts, so as to apply them more flexibly in practical problems.