Operations of Limits of Sequences

Topics: Calculus - Sequence - Limit of a Sequence

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Let $\{a_n{\}}_{n\in \mathbb{N}}$ and $\{b_n{\}}_{n\in \mathbb{N}}$ be convergent sequences. Then, their limits can be operated as follows:

1. Addition:

$$\underset{n\to \infty }{\lim }(a_n+b_n)=\underset{n\to \infty }{\lim }{a}_n+\underset{n\to \infty }{\lim }{b}_n$$

1. Product:

$$\underset{n\to \infty }{\lim }(a_n\cdot b_n)=\left(\underset{n\to \infty }{\lim }{a}_n\right)\cdot \left(\underset{n\to \infty }{\lim }{b}_n\right)$$

1. Quotient:

$$\underset{n\to \infty }{\lim }\frac{1}{a_n^{\ast }}=\frac{1}{{\lim }_{n\to \infty }{a}_n^{\ast }}$$

(when ${\lim }_{n\to \infty }{a}_n\ne 0$, and with $a_n^{\ast }$ being the subsequence of $a_n$ without any $0$ terms)