Prove that if the independent random variables \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}, \ldots\) are such that

\[ \max _{1 \leqslant k \leqslant n} \int_{|x| \geqslant A}|x| d F_{k}(x) \rightarrow 0 \text { when } A \rightarrow \infty \]

then the law of large numbers is applicable to the sequence \(\left\{\xi_{n}\right\}\).

Employ the method used in the proof of Khinchin's theorem.