$f命题1：$设$int_a^{ + infty } {fleft( x ight)dx} $收敛，若$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight)$存在，则$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight) = 0$
证明：反证法，若$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight) = l e 0$，则不妨设$l > 0$
从而 由极限的保号性知，存在$M > 0$，当$x>M$时，有$fleft( x ight) > frac{1}{2}l$，于是[mathop {lim }limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} int_M^x {fleft( t ight)dt} ge mathop {lim }limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} frac{1}{2}lleft( {x - M} ight) = + infty ] 这与$int_a^{ + infty } {fleft( x ight)dx} $收敛矛盾，故$lim limits_{x o egin{array}{*{20}{c}} {{ m{ + }}infty } end{array}} fleft( x ight) = 0$
$f{注1:}$由于$int_a^{ + infty } {fleft( x ight)dx} $收敛，所以$lim limits_{A o egin{array}{*{20}{c}} { + infty } end{array}} int_a^A {fleft( x ight)dx} $存在
$f{注2:}$若$sumlimits_{n = 1}^infty {{a_n}} $收敛，则$lim limits_{n o infty } {a_n} = 0$