65

$f命题2：$设$fleft( x ight)$在$left( {0,1} ight)$上单调,且无界广义积分$int_0^1 {fleft( x ight)dx} $收敛,则[mathop {lim }limits_{n o infty } frac{{fleft( {frac{1}{n}} ight) + fleft( {frac{2}{n}} ight) + cdots + fleft( {frac{{n - 1}}{n}} ight)}}{n} = int_0^1 {fleft( x ight)dx} ]
证明：我们不妨只讨论$fleft( x ight)$单调增加的情况,则有不等式

[int_0^{1 - frac{1}{n}} {fleft( x ight)dx} le frac{{fleft( {frac{1}{n}} ight) + fleft( {frac{2}{n}} ight) + cdots + fleft( {frac{{n - 1}}{n}} ight)}}{n} le int_{frac{1}{n}}^1 {fleft( x ight)dx} ]
令$n o infty $,则由夹逼原理即证