Question 6

Let $${x_n = \frac{(n + 1)(n + 2)}{(n - 1)(n - 2)}}$$, defined for any integer $n \ge 3$.

• Prove that the sequence $x_n$ is monotonic.
• Prove, from first principles, that $\lim_{n \rightarrow \infty} x_n = 1$.
• Also find, with proof, a real number $N$ such that $x_n$ is within $10^{-10}$ of $1$, for all integers $n > N$.