\(\huge^\star\;\underset{n\to\infty}{\lim}n=\sup\mathbb{N}\not\in\mathbb{N}\)

elim

\(\huge\color{red}{\because\quad\lim a_n=a\iff \underline{\lim}a_n=\overline{\lim}a_n=a}\)

【定理】\(\lim n=\sup \mathbb{N}\not\in\mathbb{N}\)
【证明】\(\;\because\;\underline{\lim} n=\underset{\;\;\small n\in\mathbb{N}\;n\le k\in\mathbb{N}}{\sup\;\inf} k{\small=\sup\mathbb{N}}\ge\overline{\lim} n.\)
\(\qquad\qquad\therefore\;\lim n=\sup\mathbb{N}.\)
\(\qquad\)因为\(\mathbb{N}\)无最大元, 故 \(\lim n=\sup\mathbb{N}\not\in \mathbb{N}\quad\small\square\)