Definition of the Limit and Limit Laws for Sequences

The following definition is precise, and is explained in some
detail in the video below. However, it will suffice to
intuitively consider the limit of a convergent sequence to the
number $L$ that the terms $a_n$ get closer and closer to as $n$
gets larger and larger.

Definition of the limit of a sequence

A sequence $\{a_n\}$ converges to $L$, denoted
$$\lim_{n\to\infty}a_n=L$$ if, for any number $\epsilon
>0$, there exists an integer $N$ such that $\lvert a_n
- L\rvert<\epsilon$ whenever $n>N$.

In other words, no matter how close to the limit we want
to get (within $\epsilon$ of $L$), we will eventually
(when $n>N$ for some $N$) be that close and stay that
close.

Limit Laws for Sequences

Assume that for sequences {$a_n$} and {$b_n$},
$\displaystyle\lim_{n\to\infty}a_n=L$ and
$\displaystyle\lim_{n\to\infty}b_n=M$. Then

$\displaystyle\lim_{n\to\infty}(a_n+b_n)=L+M$

$\displaystyle\lim_{n\to\infty}(a_n-b_n)=L-M$

$\displaystyle\lim_{n\to\infty}(a_nb_n)=LM$

$\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{L}{M}$
as long as $M \ne 0$, and

$\displaystyle\lim_{n\to\infty}ca_n=cL$ if $c$ is a
constant.

As a result, the sum, difference, product and ratio of two
convergent sequences automatically converge (if we're not dividing
by numbers close to zero), as do multiples of convergent
sequences. As was the case with functions, we use these limit laws to help us compute
limits of sequences.

This video explains the definition above, the limit laws, as well as
definining other important terms: bounded,
(strictly) increasing/decreasing, and monotonic sequences.