As we kown, a positive function $L: [0,+\infty)\to\mathbb{R}^+$ is called slowly varying function if $$\lim_{x\to+\infty}\frac{L(cx)}{L(x)}=1 \text{ for any $c>0$}. $$
I find the property of slowly varying function ''$\displaystyle\lim_{x\to +\infty}x^{-\delta}L(x)=0$ for any $\delta>0$" is very important. This property is published in Russian, in 1965. Can you give me its proof in English? I am very eager to know its strict mathematical proof.