Limit of a product of sequences

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. 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, $(1 + 1/n)(2 - 1/n) \to 1 \cdot 2 = 2$.