数列极限的保号性和注意事项

Posted 2020-12-19 strongdady

(【定理内容】若a_{n}leqslant b_{n},则lim_{n o infty}a_{n}leqslant lim_{n oinfty}b_{n})
【证明】
反证法。
(假设lim_{n o infty}a_{n}>lim_{n oinfty}b_{n})
(设lim_{n o infty}a_{n}=a,lim_{n oinfty}b_{n}=b)
(则quad a>b)
(设epsilon=frac{a-b}{2})
(则exists N_{1},当n>N_{1}时，|a_{n}-a|<epsilon)
(则exists N_{2},当n>N时_{2}，|b_{n}-b|<epsilon)
(设N=max{N_{1},N_{2}},则)
(当n>N时,有|a_{n}-a|<epsilon,|b_{n}-b|<epsilon)
(即quad)