A sequence of independent and identically distributed random variables (left{xi_{i} ight}) is defined by the equalities (a)

A sequence of independent and identically distributed random variables \(\left\{\xi_{i}\right\}\) is defined by the equalities

(a) \(\mathbf{P}\left\{\xi_{n}=2^{k-\log k-2 \log \log k}\right\}=\frac{1}{2^{k}}(k=1,2,3, \ldots)\)

(b) \(\mathbf{P}\left\{\xi_{n}=k\right\}=\frac{c}{k^{2} \log ^{2} k}\left(k \geqslant 2, c^{-1}=\sum_{k=2}^{\infty} \frac{1}{k^{2} \log ^{2} k}\right)\)

Prove that the law of large numbers is applicable to both sequences.