Chapter 9.1, Problem 51ES

If we accept the fact that the sequence

${\left\{\frac{n}{n+1}\right\}}_{n=1}^{+\infty }$

converges to the limit $L=1,$ then according to Definition 9.1.2, for every $\in >0$ there an integer N such that

$\left|a_n-L\right|=\left|\frac{n}{n+1}-1\right|<\in$

$\text{when}\text{}\text{}n\ge N.$ In each part, find the smallest value of N for the given value of $\in$ .

$\begin{array}{l}\left(\text{a}\right)\in =0.25\\ \left(\text{b}\right)\in =0.1\\ \left(\text{c}\right)\in =0.001\end{array}$