Arithmetic of Limits of Real Sequences

Let $(a_{n})_{n=1}^{\infty}$ and $(b_{n})_{n=1}^{\infty}$ be sequences of real numbers which converge to $A$ and to $B$ respectively, and let $c$ be a real number. Then the following hold.

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

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

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

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