---
tag: HNCY0067F1
kind: statement
role: lemma
name: Kronecker's lemma
msc: [40A05]
uses: [0CVVKY7GY2, 0AZ3R52593, 0KN22AAW5R]
sources:
  - url: https://en.wikipedia.org/wiki/Kronecker%27s_lemma
same_as:
  wikidata: Q911141
  wikipedia: Kronecker's lemma
---

## Statement

Let $(x_n)_{n \ge 1}$ be a real sequence such that the series $\sum_{n=1}^{\infty} x_n$ [[converges|0CVVKY7GY2]].
Let $(b_n)_{n \ge 1}$ be a [[monotone|0KN22AAW5R]] sequence of real numbers with $0 < b_1 \le b_2 \le \cdots$ and $b_n \to \infty$.
Then
$$\lim_{n \to \infty} \frac{1}{b_n} \sum_{k=1}^{n} b_k x_k = 0.$$

## 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 $(y_1 + \cdots + y_n)/n \to 0$.

## Explanations

### L1 human

If a series converges, its terms become small on average.
Kronecker's lemma makes this precise: even after multiplying the terms by growing weights $b_k$, the weighted sum grows more slowly than $b_n$.

### L2 human

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|0Q3AKZV74D]].