Answer
(b)
Work Step by Step
According to the limit definition:
We write $\mathop {\lim }\limits_{n \to \infty } {a_n} = L$
if, for every $\varepsilon > 0$, there is a number $M$ such that $\left| {{a_n} - L} \right| < \varepsilon $ for all $n>M$.
Thus,
$\left| {{a_n} - L} \right| < \varepsilon $,
$ - \varepsilon < {a_n} - L < \varepsilon $,
$L - \varepsilon < {a_n} < L + \varepsilon $, ${\ \ \ }$ for all $n>M$.
The last inequality is equivalent to saying the interval $\left( {L - \varepsilon ,L + \varepsilon } \right)$ contains all elements of the sequence $\left\{ {{a_n}} \right\}$ for $n>M$, except possibly some finite number of elements ${a_1},{a_2},{a_3},...,{a_M}$. In other words, stated in (b): for every $\varepsilon > 0$, the interval $\left( {L - \varepsilon ,L + \varepsilon } \right)$ contains all but at most finitely many elements of the sequence $\left\{ {{a_n}} \right\}$.
