Proving implicit function theorem with the methods of series

Implicit function theorem is one of the most classic and important results in modern mathematics[2]. It is used to describe whether one variable of a equation could be written as a function of the other variables, which also has a wide range of application[5]. For the proof of this theorem, one method is to discuss the properties of its partial derivatives in a certain field with the help of given conditions, in order to complete the continuity proof, and prove the differentiability of the function by using the differential mean value theorem. But obviously, such a proof is too complex. Therefore, in this paper, we will introduce some relevant contents of series, including the definition of series convergence and uniform convergence, the necessary conditions of convergent series, the continuity theorem of uniformly convergent function and Weierstrass discriminant, and take this as a starting point, construct a function sequence, and use the method of series to prove the implicit function theorem.