Real Analysis/Section 2 Exercises/Answers

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 Template:Math.

   $\begin{array}{cc}{a}_1& <a_2\\ \frac{1}{2}& <\frac{2}{3}\\ 3& <4\end{array}$

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

   $\begin{array}{cc}{a}_n& <a_{n+1}\\ \frac{n}{n+1}& <\frac{n+1}{n+2}\\ n(n+2)& <(n+1)(n+1)\\ n^2+2n& <n^2+2n+1\\ 0& <1\\ \end{array}$

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.