证明：由$f$可积的定义知，对任意的$varepsilon > 0$，存在$N$，使得[int_a^b {left| {fleft( x ight) - {{left[ f ight]}_N}left( x ight)} ight|dx} < frac{varepsilon }{3}]对于函数${{{left[ f ight]}_N}}$，由$f{Lusin定理}$知，对任给$delta = frac{varepsilon }{{3N + 1}} > 0$，存在闭集${E_delta } subset left[ {a,b} ight]$，以及存在$left[ {a,b} ight]$上连续函数$varphi ：$$left| {varphi left( x ight)} ight| le N$，使得$mleft( {{E_delta }} ight) < delta $，且当$x in left[ {a,b} ight]ackslash {E_delta }$时，$varphi left( x ight) = {left[ f ight]_N}left( x ight)$，于是[int_a^b {left| {{{left[ f ight]}_N}left( x ight) - varphi left( x ight)} ight|dx} = int_{{E_delta }} {left| {{{left[ f ight]}_N}left( x ight) - varphi left( x ight)} ight|dx} le frac{{2varepsilon N}}{{3N + 1}} < frac{{2varepsilon }}{3}]所以有[int_a^b {left| {fleft( x ight) - varphi left( x ight)} ight|dx} le int_a^b {left| {fleft( x ight) - {{left[ f ight]}_N}left( x ight)} ight|dx} + int_a^b {left| {{{left[ f ight]}_N}left( x ight) - varphi left( x ight)} ight|dx} < varepsilon ]