Smallest Values Which Satisfy Equation

Question

Sn = (n+1)/(n+2)

Find the smallest value of M such that $|an−1|≤0.001$ for n ≥ M

What I've tried:

$|an| ≤ 1.001$

$(n+1)/(n+2) ≤ 1.001$

$n ≤1.001n + 1.001$

$M ≤0.001n + 1.001$

if n = 1

$M ≤1.002$

I am confused by M as it is not found in the original equation.

Answer 1

\begin{align} |S_n - 1| &\le 0.001 \\ \left| \dfrac{n+1}{n+2} - 1 \right| &\le 0.001 \\ \left| \dfrac{-1}{n+2} \right| &\le 0.001 \\ \dfrac{1}{n+2} &\le \dfrac{1}{1000} \\ n+2 &\ge 1000 \\ n \ge 998 \end{align}