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Real Analysis: Math 4050-4060

Valerio De Angelis

Section 3.7 Exercises

Exercise 3.7.1.

Find possible formulas for the following sequences.
  1. \(\displaystyle (1,-1,1,-1,1,-1,\ldots)\)
  2. \(\displaystyle (0,2,0,2,0,2,\ldots) \)
  3. \(\displaystyle (0,1,0,1,0,1,\ldots)\)
  4. \(\displaystyle (a,b,a,b,a,b,\ldots)\)
Experiment by adding a constant to a known sequence, or dividing by a constant.

Exercise 3.7.3.

Retrace the steps in Example 3.2.4 to prove that \(\displaystyle \lim_{n\rightarrow \infty}a_n=\frac{3}{2}\text{.}\) That is, show that if \(n_0\) is any integer greater than \(\frac{1}{4}\left(\frac{5}{2\epsilon}+1\right)^2\text{,}\) \(n\geq n_0\) implies that \(|a_n-3/2|\lt \epsilon\text{.}\)

Exercise 3.7.4.

The hint given for the proof of Theorem 3.2.7 b. uses the identity \(a_nb_n-ab= b_n(a_n-a)+a(b_n-b)\) and Theorem 3.2.5 (a convergent sequence is bounded). Check that we can also write
\begin{equation*} a_nb_n-ab = (a_n-a)(b_n-b)+a(b_n-b)+b(a_n-a) \end{equation*}
and use this identity to give a proof that does not use Theorem 3.2.5

Exercise 3.7.5.

Prove the statement made in Subsection 3.2.5: If \(\displaystyle \lim_{n\rightarrow \infty}a_n=a\) and we define \(b_n=a_{m+n}\text{,}\) where \(m\) is a fixed positive integer, then \(\displaystyle \lim_{n\rightarrow \infty}b_n=a\text{.}\)

Exercise 3.7.6.

Prove that \(\displaystyle \lim_{n\rightarrow \infty} (-n) = -\infty \text{.}\)
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