证：只需证明充分性，设数列[tex=2.0x1.357]JcbYg1Bir/w7c3TYOky1slb0vQNA4jjXvAjG23A7ErE=[/tex]满足条件：对任给正数[tex=0.5x0.786]OpoabfWfZdF4cYFv2GsywQ==[/tex]，总存在某一个自然数[tex=0.857x1.0]HcQeTeQtUqN73yUJqDRZkQ==[/tex]，使得当[tex=4.071x1.214]Hmvk102I8Uwm1TFdgYDCzw==[/tex]时，都有[tex=5.714x1.357]0qgkGKQdX89bnCsLqHYFlGUiZ6O3ohVVNsAAE/rkP/FYQQtB483LFNXREf+P44v8[/tex]，取[tex=2.357x1.0]FxfBNiIERi60VpPUHNmU1w==[/tex]，则存在自然数[tex=1.214x1.214]TVQWqYH3jQTUZFV8bhwvGQ==[/tex]，当[tex=3.143x1.214]R0hc9aTb+pSDwdGKmrkB/g==[/tex]时，有[tex=6.857x1.357]FVW+3elqlhwuZ0nbchhQTrSCY/D001K28evGh0PJ6X+1FSd4nK940d413N/vdZ94[/tex]，从而[tex=7.357x1.357]SeiKrRHzxsNHxRVsr58lhkL9h5YBmoEW9DeLDYGStihDC3BS+B5ZJRF+PmWv/jYY[/tex]，令[tex=15.286x1.357]K951zQAeGYzl+bwlmWLZYWQV1nEcpjT7qU0LbSrK3CgPnk9jn+b6Sp9QE1Icada1239cFnNLypGYgdbLx5QDfUnSwMms249RQvz3n+tuk0htdLwxtt3X+qTplHx4OToz4RiKcZLD3Z1yPvupiSBvrA==[/tex]，则对一切[tex=5.214x1.214]r+Z9T6AUXghnBM9yfKln5g==[/tex]，有[tex=3.857x1.357]SeiKrRHzxsNHxRVsr58lhpvf44zYAkb0TJDfE2s2hpU=[/tex]，即[tex=2.0x1.357]JcbYg1Bir/w7c3TYOky1slb0vQNA4jjXvAjG23A7ErE=[/tex]有界  下证[tex=2.0x1.357]JcbYg1Bir/w7c3TYOky1slb0vQNA4jjXvAjG23A7ErE=[/tex]有收敛子列，若[tex=9.857x1.357]bspNXjPtcwP0bnLrzfhQs+gfm+PFtBoXT5GQPrWLwr98L62Nd0SU4+QvseTWUJUA[/tex]时有限集，则[tex=2.0x1.357]JcbYg1Bir/w7c3TYOky1slb0vQNA4jjXvAjG23A7ErE=[/tex]必有一常子列，若[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]为无限集，则由聚点定理，[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]有一聚点[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]，由聚点定理可证，存在[tex=2.0x1.357]qJRUlDJYGS5Hvu8eTmmu6kC8oLhWsXvaGbtVNIlY0E0=[/tex]，使[tex=5.286x1.786]lgGAiJ7xfUrmu20hqq8IlpIaXdI93C5RcxitxIuMAhNpg4z07bMSGJkeR1R2lT17[/tex]，则对任给正数[tex=0.5x0.786]OpoabfWfZdF4cYFv2GsywQ==[/tex]，存在[tex=0.857x1.0]HcQeTeQtUqN73yUJqDRZkQ==[/tex]，当[tex=5.071x1.214]jyB6F0aLrwDc/MrRxU6q1g==[/tex]时，[tex=6.0x2.143]qhwohtWhiMlH+sRrCW5r2+JYnlfeSttEg8u6oCYTDks/qV3DvIp17J8JR8TXfmcd8aVUgV9frZ4evs/JIlzTNQ==[/tex]，[tex=5.857x2.143]N+rnslg9M6txMSTSWwBDT85XbYFUMoVbDlcsM4a6kntn1f8L1tRh9AmbwH6gwby1[/tex]，所以[tex=2.786x1.071]N7hi+AppX8MZXT7sGYf3Yw==[/tex](任取[tex=3.0x1.214]isewyc7Oys/ZUCPyOh6cCA==[/tex] 使[tex=2.929x1.071]MoweMg6gAUBxPjBCQqs38g==[/tex]）时，有[tex=19.286x2.143]QMvizReWVwcZlQWlKuCD3fZ5ke6VcGERDDMI+DbudKf9hxU06tzeS9jsVp7GH1a+JNpfPV9QTHs8yM/bdkKFz5NXU5v7farDvjHCiNfKByHJYdeMOdnUMrcNe0OJbuDu5sK1Y105jn6yLfXNYdkL76hZKFfkGXhbbhl9QFJL+9dMrYf3n1l7FtnulVMVr+0u[/tex]故[tex=5.0x1.786]lgGAiJ7xfUrmu20hqq8IlpIaXdI93C5RcxitxIuMAhM7Nt0s0Q671LRkNCHI19mJ[/tex]