Limit Algebra for Sequences

Limit algebra for sequences: Let $(a_n)$ and $(b_n)$ be real sequences, and suppose $a_n \to a$ and $b_n \to b$ (in the sense of a convergent sequence ). Then:

• $a_n+b_n \to a+b$ and $a_n-b_n \to a-b$.
• For any $c\in\mathbb{R}$, $c\,a_n \to c\,a$.
• $a_n b_n \to ab$.
• If $b\ne 0$ and $b_n\ne 0$ for all sufficiently large $n$, then $\frac{a_n}{b_n}\to \frac{a}{b}$.
• $|a_n|\to |a|$ (see absolute value ).

These rules are used constantly to build new limits from known ones and often pair with the squeeze theorem for estimates.