自然数集是无限集 反证法 无限变有限，构造鸽子与笼子 Suppose to the contrary that N is finite.Since N0 there exists an integer m and a one-to-one mapping,g,of N onto (1,2,...,m).Now (1,2,...,m+1)N,so we may consider the restriction gl.2..m+: (1,2,...,m+1)(1,2,...,m).The pigeonhole principle(Theorem 21.2)implies that gis not one-to-one.This,in turn,implies (as you surely showed in Exercise 20.9)that g is not one-to-one, contradicting our choice of g.Therefore,it must be the case that N is infinite. ■自然数集是无限集 反证法 无限变有限，构造鸽子与笼子