用数学归纳法:
(1)当n=2时,左边=1=右边
(2)假设当n=k时,有1×1!+2×2!+3×3!+...+(k-1)×(k-1)!=k!-1
则当n=k+1时,
1×1!+2×2!+3×3!+...+(k-1)×(k-1)!+k*k!
=k!-1 +k*k!
=(k+1)k!-1
=(k+1)!-1
所以,当n=k+1时,命题成立
综上,原命题成立