91Ó°ÊÓ

For each of the following statements, determine whether it is true or false and justify your answer. a. Every bounded sequence converges. b. A convergent sequence of positive numbers has a positive limit. c. The sequence \(\left\\{n^{2}+1\right\\}\) converges. d. A convergent sequence of rational numbers has a rational limit. e. The limit of a convergent sequence in the interval \((a, b)\) also belongs to \((a, b)\).

Which of the following sequences is monotone? Justify your conclusions. a. \(\left\\{n+\frac{(-1)^{n}}{n}\right\\}\) b. \(\left\\{\frac{1}{n^{2}}+\frac{(-1)^{n}}{3^{n}}\right\\}\)

Show that the set \((-\infty, 0]\) is closed.

For each of the following statements, determine whether it is true or false and justify your answer. a. The set of irrational numbers is closed. b. The set of rational numbers in the interval [0,1] is compact. c. The set of negative numbers is closed.

Using only the Archimedean Property of \(\mathbb{R},\) give a direct \(\epsilon-N\) verification of the following limits: $$\text { a. } \lim _{n \rightarrow \infty} \frac{1}{\sqrt{n}}=0 \quad \text { b. } \quad \lim _{n \rightarrow \infty} \frac{1}{n+5}=0$$

Show that a strictly increasing sequence has no peak indices.

Suppose that the sequence \(\left\\{a_{n}\right\\}\) converges to \(a\) and that \(a>0 .\) Show that there is an index \(N\) such that \(a_{n}>0\) for all indices \(n \geq N\).

Show that for a monotonically decreasing sequence every index is a peak index.

Suppose that the sequence \(\left\\{a_{n}\right\\}\) converges to \(\ell\) and that the sequence \(\left\\{b_{n}\right\\}\) has the property that there is an index \(N\) such that $$a_{n}=b_{n}$$ for all indices \(n \geq N\). Show that \(\left\\{b_{n}\right\\}\) also converges to \(\ell .\) (Suggestion: Use the Comparison Lemma for a quick proof.)

Prove that the Archimedean Property of \(\mathbb{R}\) is equivalent to the fact that \(\lim _{n \rightarrow \infty} 1 / n=0\).