证：设原收敛级数为[tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex]，当然有[tex=4.214x1.714]OqU0SQaVHd2x+OGLCy0gvatsU7KHTyxR18qBsaLaMzD5vjob45CraI1GSpzkBzS9[/tex]。又记重排出的新级数为[tex=2.5x3.286]3PXegz5bAQsuTODB0U8KrGvHGhSZgIwzLu/Z7gPL7V8=[/tex]，再记[tex=2.571x3.286]3PXegz5bAQsuTODB0U8KrMk2INBod3J4j6UNzA1luB4=[/tex]的前[tex=0.857x1.0]+NBI8Pm2vVS+bGgOpHKyOA==[/tex]项部分和为[tex=5.286x2.857]LVy4zvs8dsYtSeOe5OyNaSXebO02s9tPMTpl3YmS8dI=[/tex]，记[tex=2.5x3.286]3PXegz5bAQsuTODB0U8KrGvHGhSZgIwzLu/Z7gPL7V8=[/tex]的前[tex=0.857x1.0]+NBI8Pm2vVS+bGgOpHKyOA==[/tex]项部分和为[tex=5.143x2.857]NX1hxrLqABDtGxwI2u3HkKNSAXS9dSnbASQS4hevCTaxmVypUfP1iMhTJejrU93L[/tex]，当然有[tex=5.714x1.571]svJWG39FuHsdckNRem17DjfB2AiCW2UwSwdKBVX0/Kd87kUQ6VxDAvkx6RLhJcLN[/tex]。 今证[tex=1.143x1.0]nWk/d8F3tDaODV5Gf5xqdQ==[/tex]的极限也存在，且等于[tex=0.643x1.0]VuDqnB7C7a0HJjCNT6LA5A==[/tex]。考察[tex=1.143x1.0]nWk/d8F3tDaODV5Gf5xqdQ==[/tex]与[tex=1.214x1.214]F/bIXRrMD4Ib9Rm8CSwQFA==[/tex]之差[tex=5.286x1.214]87I1OckKc4ruNlAe6GZGa2z3bgSgZtdvAZBgXk1X2QTT3xFpEY94bPjhLM9ZnlTY[/tex]，任给[tex=2.357x1.071]zaTYmiB02c3fW3zvAQdizg==[/tex]，取[tex=5.214x2.143]M77Im89n9ijU205Hut5rnilDsFPcQPpu3mtFWShxj619g7TvbmO6+vKnVhcWRQiD[/tex]，则存在[tex=1.214x1.214]FKy8/D8RYfNMeysc62gHRA==[/tex]，使当[tex=3.143x1.214]lxNsqwp1sZGa+M7mJ8/6bkgL85X7Cb1fCQfixUVfZBo=[/tex]时，有[tex=3.143x1.357]P6xogcT0v+MaB5vfTP/QuUJvABkRBAmELMFVHEP5DEDuiSkBQcGmvuJjVAxXwDeW[/tex]，今取[tex=5.5x1.214]q3AF9g8OAGPlyWMYc0M57ThZGzYVsvsExTmMprEJ+vU=[/tex]，又记[tex=1.0x1.214]3cqhww/dYMlkOsXBcaexuA==[/tex]内各[tex=1.0x1.0]/DJc0lEQ/Y1auXDMJlAodQ==[/tex]项元素集合为[tex=1.071x1.5]NyBiPdSBQJ90uej0anjadZhdWjCcOxCK9X+JQBN8Gh0=[/tex]，记[tex=1.0x1.0]gjPmOjG4xi1eTWq5rIOeeA==[/tex]内各 $b_{n}$ 项元索集合为[tex=1.0x1.286]JkVHvovn4dd+OjhbwDbtIc8rMEyX/WE28JwvcOhCdG0=[/tex]，则有[tex=11.214x2.5]EFowu6n625szdNT6705VomjDlIrfkF2OjdId9e1BoXui9UZz0cE6zbZjMw5Ev4r7GxIvxWKCuHQ/Z5ucD8On10PiuKKxC8h+w4iENXgUazX0zKbqHmUldi+4wRz9co4BqOyecy+L2us7CZBeqHGLmg==[/tex]，今从[tex=0.929x1.0]iFE0cCPkws2/Jm5e9iHeuw==[/tex]查起，看[tex=4.071x1.0]DkQMvCDF/4vyPYjHN/R9lSXzISxWx1S1fgfu39r3iBA=[/tex]至[tex=1.143x1.0]k/ec5EQhFarvOC2ZFfZCpA==[/tex]，注意每一个[tex=0.786x1.0]k2s61G+KZI8Yw0ah0O7QEQ==[/tex]被重排成[tex=0.786x1.286]hU32ZcKTHzROpgZYN8+tgQ==[/tex]时，[tex=0.357x1.0]O88k7AtkDgTC9kv/8dY0lg==[/tex]与[tex=0.429x1.214]rmIPPJrP+tFN2kAYPlU/4g==[/tex]的标号差不超过[tex=0.929x0.786]FTfUoplPStit3eMYfNbP0g==[/tex]。因此，对每一 个[tex=0.786x1.0]k2s61G+KZI8Yw0ah0O7QEQ==[/tex]总可以在[tex=0.714x1.214]DoNFZRjsvUBa9i6miU5BKg==[/tex]的前后各不超过[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]个元素内找到一个[tex=2.286x1.286]X/IagupgsDQBuBma2JoKUg==[/tex]。反过来，从[tex=0.786x1.214]Yc8v8Kxc8MA1NQozZkQtNg==[/tex]查起，看[tex=3.857x1.214]DZ/hADVhu6U78WfIzHYjKMIrpzq9g5z0ga1PKrPMrHo=[/tex]至[tex=1.0x1.214]LDTku/ptS77a4tU99AjmbA==[/tex]，对每一个[tex=0.429x1.0]Q2fWySASH/4Xf2eu85OwAQ==[/tex]，总可以在[tex=0.857x1.071]jnTrYz1KWSsAGRcZmwz9kg==[/tex]的前后各不超过[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]个元素内找到一个[tex=2.286x1.214]Ebea5t3ChzTf5WRMGzAJ5g==[/tex]。但也可能且只有那种可能：最后一段不超过个[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]元素的[tex=0.786x1.0]k2s61G+KZI8Yw0ah0O7QEQ==[/tex]，即[tex=8.286x1.0]s1muBzHJJQowW62OTL0Wn0JPeSr8K/FDF8hh2yq6otA8tqAWLbeZJnADsI1EaaR4[/tex]之内若干个元素可能被迁到[tex=1.0x1.214]LDTku/ptS77a4tU99AjmbA==[/tex]之后，从而，在 [tex=1.143x1.286]JkVHvovn4dd+OjhbwDbtIbmhI+Hr/7coBhvSbRFOK70=[/tex]内找不到搬迁元素，但个数(设为[tex=0.5x0.786]U5O66aolbR1y5vuKrQbXNA==[/tex]个)不超过[tex=0.929x0.786]FTfUoplPStit3eMYfNbP0g==[/tex]。同样，也有可能最后一段不超过[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]个元素的[tex=0.786x1.286]hU32ZcKTHzROpgZYN8+tgQ==[/tex]，即[tex=7.929x1.214]uXG2mzGuia7j8K6dGrIJsbOJRDf//q6mAJtyj8JOlFSIIkjv9AKTjAYVufaE5r8U[/tex]之内若干个元素在[tex=1.143x1.5]NyBiPdSBQJ90uej0anjadRZ+mvh236yTbc0w6Y0CZ1Y=[/tex]内找不到搬迁元素，但个数(设为[tex=0.5x0.786]BgHR5DBWke5rTEC5XEckiQ==[/tex]个)不超过[tex=0.929x0.786]FTfUoplPStit3eMYfNbP0g==[/tex]。除此之外均有对应的搬迁元素且一一对应。于是，[tex=36.429x5.929]u3KvxPvA1dJyDrT0dQHwL8YpGYRi5fgWVaowe44WQODIfsA/X5gduzTu6e4eQugOsajD1ALFBQG2jEFsraQrQEE7/zPaV/rurfg/AF7BtZ18mOueG6d0qD3yGEdE1Ei/Dhc0nmzR3mjQ/w7rFTKaDcDUGu5xNXN53pKxKkJVrSkZg3lCp0/9mr8WsYLx4sFYrsFxOX4Pt/YAEnAa/u38kMeA9IYCfefmCzdy73YYMRoBJf6kmiajH25CrLf2JGmFfDrvhBhgKo1adUbHmYkF1cHGtUNI7R/xDDL6wXIUkO2F02TxqyCYyZwjWow35w6tlWDKa0czZKRX1xxaC7693fOy3YSXXl1L1NXS2bc4Y87+WGEnUGoHxHlz37br0jJ4pTF2DdhvxJCkIXjThTR+pngV4NPV0PH/dECj/rjcNJGBZW6MRTHL8GToS3ULSjNNEm1DNDdK7pIscaq5H3Z2HJ0KRBVihwcw4DQufl/ZB926bYE+UCqLY3hLUQw9P8wv0tJwwfbntwjS+BlsY36WdEomvbyYSuVDeHT0JUPvXWSF6i+PgjDNru8grL/Nmu5qvsUPY14K1wtDEC+bzzykTBSyKgAqnzItDbpoMTCWiq8TPJWmi4GUylRjo6Vsvfrf7LAOCguSAafIIPtEx14H1+sfUqT6pIDMmiQCfaSLPSs=[/tex]上式中[tex=1.0x1.0]/DJc0lEQ/Y1auXDMJlAodQ==[/tex]的下标[tex=7.286x1.214]lxNsqwp1sZGa+M7mJ8/6bscPDXjifTaW/8WMp2VotXs=[/tex]，故[tex=3.357x1.357]P6xogcT0v+MaB5vfTP/QubV+oeghY2L6rzyp+xzlkM4=[/tex]。而[tex=0.857x1.214]vTBgjtz1X521bRmrLanE/Q==[/tex]的下标[tex=4.786x1.214]WAI0XiyPnayfiWDromftk3J1JTwH3+F143+/IkCEAl0=[/tex]，记住[tex=0.857x1.214]vTBgjtz1X521bRmrLanE/Q==[/tex]由某[tex=0.786x1.0]k2s61G+KZI8Yw0ah0O7QEQ==[/tex]搬迁而来，其下标在[tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]的前后距离不超过[tex=0.929x0.786]VF0GLe2VBE/4VKNzpyOfFg==[/tex]，故此时[tex=2.857x1.214]YVlq+XqsfN8l/WJDE0/++dBxmkX4X/0r/1NWgRl+Su0=[/tex]，因而，此时[tex=5.5x1.357]Tg6ngJ4g2/S0oiALGf6cV66vZaxEAO+paEVLkrmBl1wZ/mxJy7BWrZTG+9tjZtX/IoKU+myCHB/O6OSWZYF8tg==[/tex]。从而上述不等式是成立的。由极限定义知[tex=4.857x1.786]qoopFZdUXJ3oB3cj+yEEXSY4MHAA4VrRUZTgjUBzQprh4tssyl3C6f+uzkRkDwbM[/tex]，也即有[tex=9.929x1.786]jI0NthCJ2pXG96E9XXkzEYzEWLIARChC7SNTT1G5375Ym9PSrRE8MegIOhA0JWYj9EYe0mpPIQ+kiOeXUxQpZ8Wrv6VqOOgXPlg8A2pOntmKnRyWPk5RUfU8cU4q6SHL[/tex]。从而，命题得证。