首先由f(x)单调减,及∫{a,+∞} f(x)dx收敛,有f(x) ≥ 0.
根据Cauchy收敛准则,易得lim{x → +∞} ∫{x/2,x} f(t)dt = 0.
又f(x)单调递减,∫{x/2,x} f(t)dt ≥ ∫{x/2,x} f(x)dt = x·f(x)/2.
于是0 ≤ lim{x → +∞} x·f(x) ≤ 2·lim{x → +∞} ∫{x/2,x} f(t)dt = 0.
即lim{x → +∞} x·f(x) = 0.