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.

Let $x_1, x_2, . . .$ and $0 < b_1 < b_2 < · · · $ be sequences of real
numbers such that $\{b_n\}$ increases to infinity as $n → ∞.$ Suppose that the sum
$\sum_{n=1}^\infty\frac{x_n}{b_n}$ converges to a finite limit. Then,

$\frac{x_1+\cdots+x_n}{b_n}\to 0$ as $n\to \infty.$

Remark: It can be applied to the proof of strong law of large number.