Real Analysis/Section 2 Exercises/Answers

Problem 1
Strictly Increasing: \begin{align} a_1 &< a_2 \\ \frac{1}{2} &< \frac{2}{3} \\ 3 &< 4 \end{align} $$ \begin{align} a_n &< a_{n+1} \\ \frac{n}{n+1} &< \frac{n+1}{n+2} \\ n(n+2) &< (n+1)(n+1) \\ n^2+2n &< n^2 + 2n + 1 \\ 0 &< 1 \\ \end{align} $$
 * 1) We need to establish if it's monotone, and what kind (is it strictly increasing, strictly decreasing, non-increasing, non-decreasing, what?). Given the problem, we'll assume strictly increasing.
 * 2) First, we should prove the base case. This means proving that $a_{1} < a_{2}$. $$
 * 1) Next, we should prove that this works for any number n $$
 * 1) You're done! Everything checks out and is valid.

General
For problem 2, we see that supx_n =1 since  n/(n+1) gets close to 1.