Let $(a_{n})_{n=1}^{\infty}$ and $(b_{n})_{n=1}^{\infty}$ be \reftext{def:sequence-in-set-2026a}{sequences} of \reftext{def:real-numbers-c54-2026c}{real numbers} which \reftext{def:limit-sequence-real-c54-2026a}{converge} to $A$ and to $B$ respectively, and let $c$ be a real number. Then the following hold.

\textbf{1. (Sums)} The sequence $(a_{n}+b_{n})$ converges to $A+B$.

\textbf{2. (Products)} The sequence $(a_{n}b_{n})$ converges to $AB$.

\textbf{3. (Scalar multiples)} The sequence $(ca_{n})$ converges to $cA$; in particular $(-a_{n})$ converges to $-A$ and $(a_{n}-b_{n})$ converges to $A-B$.

\textbf{4. (Quotients)} If $B\neq0$ and $b_{n}\neq0$ for every $n$, then the sequence $(a_{n}/b_{n})$ converges to $A/B$.