---
tag: 0734RSVF2V
kind: statement
role: proposition
name: Limit of a product of sequences
msc: [40A05]
uses: [0XKWSSY8P2, 0S4JF619V7]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Properties
---

## Statement

Let $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.
If $a_n \to a$ and $b_n \to b$ as $n \to \infty$, then $a_n b_n \to a b$ as $n \to \infty$.

## Remarks

The proof uses the identity $a_n b_n - a b = a_n (b_n - b) + b (a_n - a)$. The sequence $(a_n)$ is bounded, because [[convergent sequences are bounded|0S4JF619V7]].
With a constant sequence $a_n = c$, it gives $c\, b_n \to c\, b$.
By induction, it gives $a_n^k \to a^k$ for each natural number $k$.

## Explanations

### L1 human

We may multiply the limits to get the limit of the product.
For example, $(1 + 1/n)(2 - 1/n) \to 1 \cdot 2 = 2$.