数学归纳法是论证与自然数n有关的命题的一种常用思想方法,应用范围极宽,它在数学竞赛中占有特殊地位。如何合理、灵活地运用数学归纳法?这是下面所要谈及的主要问题。一、因势利导——善“退”完成数学归纳法,两个基本步骤缺一不可。比较而言,使我们陷入困境的多数归于递推步然而纵观递推步完成的百般变化,归结起来,首要的是善于因势利导,由“k+1”退到“k”。例1 证明:对任何非空的有限集合,都可以把它的所有子集排成一列,使得除第一个外,每一个子集合都可以由它前面的那个子集增加或减少一个元 Mathematical induction is a commonly used method of thinking that propositions related to the natural number n have a wide range of applications and it occupies a special position in the mathematical competition. How to use mathematical induction rationally and flexibly? This is the main issue to be discussed below. First, take advantage of the trend - good “retreat” to complete the mathematical induction, two basic steps are indispensable. In comparison, the majority of our difficulties are attributed to recursive steps. However, looking at the various changes that are made by recursive steps, in the first place, it is good that we are good at taking advantage of the trend and retreating from “k+1” to “k”. Example 1 proves that for any non-empty finite set, all its subsets can be arranged in a row, so that except for the first one, each subset can be increased or decreased by one yuan from the previous subset.