\(\huge\color{red}{用冯氏定义证明：若\lim n\notin\mathbb{N},则\mathbb{N}=\phi}\)

春风晚霞

       由冯\(\cdot\)诺依曼自然数定义（或自然数生成法则）得：\(0=\phi\)，\(1=\{0\}\)，\(2=\{0，\)\(1\}\)，\(3=\{0，1，2\}\)，…，\(k=\{0，1，2，…（k-1)\}\),…  .
       两端分别求并得:\(\displaystyle\bigcup_{k=0}^{\infty}k=\)\(\displaystyle\bigcup_{k=1}^{\infty}\{0,1,2,3,…,(k-1)\}\) .所以\(\{1,2,3,…,\displaystyle\lim_{n \to \infty}n\}=\)\(\displaystyle\bigcup_{n=1}^{\infty}\{0,\)\(1,2,3,…,\)\((n-1)\}\).
       若\(\displaystyle\lim_{n \to \infty}n\notin\mathbb{N}\),则\(\mathbb{N}\cap\displaystyle\lim_{n \to \infty}\{1,2,…n\}=\)
\( \mathbb{N}\cap\displaystyle\bigcup_{n=1}^{\infty}\{0,1,2,3,…,(n-1)\}=\)\( \displaystyle\bigcup_{n=1}^{\infty}\mathbb{N}\)\(\cap\{0,1,2,\)\(3,…,(n-1)\}=\phi\).所以\(\mathbb{N}\cap\{0\}=\)\(\mathbb{N}\cap\{0,1\}=\)\(\mathbb{N}\cap\{0,1,2\}=\)……\(=\mathbb{N}\cap\{0,1,2，…(\displaystyle\lim_{n \to \infty}n-1)\}=\phi\),所以\(\mathbb{N}=\phi\)