1.
级数:u1v1+u1v2+u2v1+...+u1vn+...收敛且其和为w
->柯西乘积u1v1+(u1v2+u2v1)+...+(u1vn+u2vn-1+...unv1)+... 收敛,且其和为w
2.
级数u1v1+u1v2+u2v1+...+u1vn+...绝对收敛,即
|u1v1|+|u1v2|+|u2v1|+...+|u1vn|+...收敛;
->|u1v1|+(|u1v2|+|u2v1|)+...+(|u1vn|+|u2vn-1|+...|unv1|)+... 收敛;
又|u1v1|+(|u1v2|+|u2v1|)+...+(|u1vn|+|u2vn-1|+...|unv1|)+... 各项>=|u1v1|+|u1v2+u2v1|+...+|u1vn+u2vn-1+...unv1|+...各项
于是级数|u1v1|+|u1v2+u2v1|+...+|u1vn+u2vn-1+...unv1|+...收敛
即柯西乘积u1v1+(u1v2+u2v1)+...+(u1vn+u2vn-1+...unv1)+... 绝对收敛.
综上所述:柯西乘积级数也绝对收敛且其和为w.