Kronecker's lemma

Remarks

Let $r_k = \sum_{j > k} x_j$ be the tails of the series, so $x_k = r_{k-1} - r_k$ with $r_k \to 0$. Summation by parts writes $\frac{1}{b_n} \sum_{k \le n} b_k x_k$ as $b_1 r_0 / b_n - r_n$ plus a weighted average of $r_1, \ldots, r_{n-1}$ with weights $(b_{k+1} - b_k)/b_n$. These weights are nonnegative, their sum is at most $1$, and each fixed weight tends to $0$, so the average tends to $0$ because $r_k \to 0$. Example: with $b_n = n$, if $\sum y_n / n$ converges, then $(y_1 + \cdots + y_n)/n \to 0$.

Going deeper

For a student

The main use is in probability. To prove a strong law of large numbers, one shows that $\sum X_n / n$ converges almost surely, and Kronecker's lemma with $b_n = n$ then gives $(X_1 + \cdots + X_n)/n \to 0$. The proof is a summation by parts followed by an averaging argument, as in the Stolz–Cesàro theorem.