A result from calculus is Kronecker's Lemma, which states that if \(\left\{b_{n}ight\}_{n=1}^{\infty}\) is a monotonically increasing sequence of real numbers such that \(b_{n} ightarrow \infty\) as \(n ightarrow \infty\), then the convergence of the series

\[\sum_{n=1}^{\infty} b_{n} x_{n}\]

implies that

\[\lim _{n ightarrow \infty} b_{n}^{-1} \sum_{k=1}^{n} x_{k}=0\]

Use Kronecker's Lemma to prove the second result in Corollary 3.1. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(E\left(X_{n}ight)=0\) for all \(n \in \mathbb{N}\). Prove that if

\[\sum_{n=1}^{\infty} b_{n}^{-2} E\left(X_{n}^{2}ight)<\infty\]

then \(b_{n}^{-1} S_{n} \xrightarrow{\text { a.c. }} 0\) as \(n ightarrow \infty\).