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本帖最后由 elim 于 2024-5-6 20:50 编辑
定义:对\(\delta>0,\,a\in\mathbb{R},\)称\(N(a,\delta)=(a-\delta,a+\delta)\)为\(a\)的\(\delta\)-邻域.
命题0:\(\displaystyle\lim_{n\to\infty}a_n=a\iff \exists a\,\forall \delta>0\,(\{n\in\mathbb{N}: a_n\not\in N(a,\delta)\}\text{是有限集}.\)
证:\(\big(\exists a\,\forall\varepsilon>0\exists N\forall n>N\,(|a_n-a|< \varepsilon)\big)\iff\)
\(\quad\big(\exists a\,\forall\varepsilon>0\exists N\forall n>N\,(a_n\in N(a, \varepsilon)\big)\iff\)
\(\quad\big(\exists a\,\forall\varepsilon>0\exists N\,(a_n\not\in N(a, \varepsilon)\implies n\le N\big)\iff\)
\(\quad\big(\exists a\,\forall \delta>0\,(\{n\in\mathbb{N}: a_n\not\in N(a,\delta)\}\text{是有限集}.\big)\)
定义\((^\star)\)若对任意\(\delta>0,\;\{n\in\mathbb{N}^+: a_n\not\in N(a,\delta)\}\)是有限集,则称\(\displaystyle\lim_{n\to\infty}a_n=a\).
注记:据命题0, 定义\((^\star)\)与序列极限的\(\varepsilon\)-\(N\)定义(Weierstrass)是等价的, 但前者
\(\quad\)更清晰的表明极限是收敛序列的内在(固有)属性, 不以\(\small n\to\infty\)与否为
\(\quad\)转移. \(n\to\infty\)也不保证极限属于数列的值域.
例1:\(\displaystyle\lim_{n\to\infty}{{\small\frac{1}{n}}=0\not\in\big\{\small\frac{1}{n}\mid n\in\mathbb{N}^+}\big\}\) (\(\{\frac{1}{n}\}\) 以不属于其值域的\(0\)为极限.
例2:设\(f\)在含\(a\)的开集上可微, 则 \(a_n{\small=\dfrac{f(a+\frac{1}{n})-f(a)}{\frac{1}{n}}}\to \small f'(a)\;(n\to\infty).\)
\(\quad\)但没有n 使\(\frac{1}{n}=0\). 所以极限的Weierstrass 定义化解了第二次数学危机:
\(\quad\)求导不涉及以0为分母的操作,差商的极限不是终极商, 而是差商序列的聚点. |
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