> [!NOTE] **Lemma** > Suppose we have two [[Convergence|convergent sequences]] $a_{n} \to a$ and $b_{n} \to b$. Then > > (1) [[Sum Rule For Limits of Real Sequences]]: $a_{n} +b_{n} \to a+b$ > > (2) [[Product Rule For Limits of Real Sequences]]: $a_{n}b_{n} \to ab$ > > (3) [[Quotient Rule For Limits of Real Sequences]]: if $b \neq 0$ then $\frac{a_{n}}{b_{n}} \to \frac{a}{b}$ (*quotient rule*). **Proof**: The proofs all make use of [[Absolute Value Satisfies Triangle Inequality]]. # Applications See [[Algebra of Limits of Real Functions]].