Kronecker’s lemma

Kronecker’s lemma gives a condition for convergence of partial sums of real numbers, and for example can be used in the proof of Kolmogorov’s strong law of large numbers.

Lemma (Kronecker).

Let $x_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }$ and be sequences of real numbers such that $b_n$ increases to infinity as $n\mathrm{\to }\mathrm{\infty }$. Suppose that the sum ${\mathrm{\sum }}_{n\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{x}_n\mathrm{/}{b}_n$ converges to a finite limit. Then, $b_n^{\mathrm{-}\mathrm{1}}\mathit{}{\mathrm{\sum }}_{k\mathrm{=}\mathrm{1}}^n{x}_k\mathrm{\to }\mathrm{0}$ as $n\mathrm{\to }\mathrm{\infty }$.

Proof.

Set $u_n={\sum }_{k=1}^n{x}_k/b_k$, so that the limit $u_{\mathrm{\infty }}={lim}_{n\to \mathrm{\infty }}{u}_n$ exists.Also set $a_n={\sum }_{k=1}^{n-1}(b_{k+1}-b_k)u_k$ so that

$$\frac{a_{n+1}-a_n}{b_{n+1}-b_n}=u_n\to u_{\mathrm{\infty }}$$

as $n\to \mathrm{\infty }$. Then, the Stolz-Cesaro theorem says that $a_n/b_n$ also converges to $u_{\mathrm{\infty }}$, so

$$b_n^{-1}\sum _{k=1}^n{x}_k=b_n^{-1}\sum _{k=1}^n{b}_k(u_k-u_{k-1})=u_n-b_n^{-1}{a}_n\to 0.$$

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