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 a₁ < a₂.

   ${\begin{aligned}a_{1}&<a_{2}\\{\frac {1}{2}}&<{\frac {2}{3}}\\3&<4\end{aligned}}$

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

   ${\begin{aligned}a_{n}&<a_{n+1}\\{\frac {n}{n+1}}&<{\frac {n+1}{n+2}}\\n(n+2)&<(n+1)(n+1)\\n^{2}+2n&<n^{2}+2n+1\\0&<1\\\end{aligned}}$

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.