Kronecker's lemma

In mathematics, Kronecker's lemma (see, e.g., Shiryaev (1996, Lemma IV.3.2)) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the German mathematician Leopold Kronecker.

If ${\displaystyle (x_{n})_{n=1}^{\infty }}$ is an infinite sequence of real numbers such that

exists and is finite, then we have for all ${\displaystyle 0<b_{1}\leq b_{2}\leq b_{3}\leq \ldots }$ and ${\displaystyle b_{n}\to \infty }$ that

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