Definition: A sequence $\{a_n\}$ converges to $L$ 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 ($\epsilon$-close), we will eventually get and stay there, where "eventually" means "after $N$ steps".

Limit Laws

The limit laws for sequences are almost exactly the same as the limit laws for functions. For instance, if the sequence {$a_n$} converges to $L$ and the sequence {$b_n$} converges to $M$, then

$\{a_n+b_n\}$ converges to $L+M$

$\{a_n-b_n\}$ converges to $L-M$

$\{a_nb_n\}$ converges to $LM$

$\displaystyle\left\{\frac{a_n}{b_n}\right\}$ converges to $\displaystyle\frac{L}{M}$ as long as $M \ne 0$, and

If $c$ is a constant, $\{ca_n\}$ converges to $cL$.

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.