Real Analysis/Section 2 Exercises/Answers

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On Sequences[edit]

Problem 1[edit]

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[edit]

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