## On Sequences

### Problem 1

Strictly Increasing:

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 a1 < a2.

\begin{align} a_1 &< a_2 \\ \frac{1}{2} &< \frac{2}{3} \\ 3 &< 4 \end{align}

3. Next, we should prove that this works for any number n

\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}

4. 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.