Chapter 2

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 \((a, b)\).

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.

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.

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

Let \(a\) and \(b\) be numbers with \(a

Using only the Archimedean Property of \(\mathbb{R},\) give a direct \(\epsilon-N\) verification of the convergence of the following sequences: a. \(\left\\{\frac{2}{\sqrt{n}}+\frac{1}{n}+3\right\\}\) b. \(\left\\{\frac{n^{2}}{n^{2}+n}\right\\}\)

Let \(S\) be the set of rational numbers in the interval [0,2] . a. Using the definition of sequential compactness, show that \(S\) is not sequentially compact. b. Using the definition of compactness, show that \(S\) is not compact. c. Using the definition of closedness, show that \(S\) is not closed.

Suppose that the sequence \(\left\\{a_{n}\right\\}\) converges to \(a\) and that \(|a|<1 .\) Prove that the sequence \(\left\\{\left(a_{n}\right)^{n}\right\\}\) converges to 0.

Let \(S\) be a set consisting of a single point. Show that \(S\) is compact

Show that a strictly increasing sequence has no peak indices.

Show that a sequence \(\left\\{a_{n}\right\\}\) is bounded if and only if there is an interval \([c, d]\) such that \(\left\\{a_{n}\right\\}\) is a sequence in \([c, d]\).

Let \(S=[0,1] \cup[3,4] .\) Show that the set \(S\) is compact.

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

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 \(n \geq N\).

Let \(A\) and \(B\) be compact sets. Show that the union \(A \cup B\) and the intersection \(A \cap B\) are also compact.

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.)

Let \(\left\\{b_{n}\right\\}\) be a bounded sequence of nonnegative numbers and \(r\) be any number such that \(0 \leq r<1 .\) Define $$ s_{n}=b_{1} r+b_{2} r^{2}+\cdots+b_{n} r^{n} $$ for every index \(n\). Use the Monotone Convergence Theorem to prove that the series \(\left\\{s_{n}\right\\}\) converges.

Prove that the sequence \(\left\\{c_{n}\right\\}\) converges to \(c\) if and only if the sequence \(\left\\{c_{n}-c\right\\}\) converges to 0 .

For each natural number \(n\), let \(a_{n}\) and \(b_{n}\) be numbers such that \(a_{n}

A sequence \(\left\\{a_{n}\right\\}\) was defined to be bounded provided that there is a number \(M\) such that \(\left|a_{n}\right| \leq M \quad\) for every index \(n\) Show that \(\left\\{a_{n}\right\\}\) is bounded if and only if there are numbers \(a\) and \(b\) with \(a

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

For a pair of positive numbers \(\alpha\) and \(\beta,\) the number \(\sqrt{\alpha \beta}\) is called the geometric mean of \(\alpha\) and \(\beta,\) and the number \((\alpha+\beta) / 2\) is called the arithmetic mean of \(\alpha\) and \(\beta\). By observing that \((\sqrt{\alpha}-\sqrt{\beta})^{2} \geq 0,\) show that \((\alpha+\beta) / 2 \geq \sqrt{\alpha \beta}\).

For each natural number \(n,\) let \(I_{n}\) be a closed bounded interval. Suppose that \(\left\\{I_{n}\right\\}_{n=1}^{\infty}\) covers the compact set consisting of the closed bounded interval [0,1] . Is it true that this cover has a finite subcover?

Prove that a sequence \(\left\\{a_{n}\right\\}\) does not converge to the number \(a\) if and only if there is some \(\epsilon>0\) and a subsequence \(\left\\{a_{n_{k}}\right\\}\) such that $$ \left|a_{n_{k}}-a\right| \geq \epsilon $$ for every index \(k\).

Prove that $$\lim _{n \rightarrow \infty} n^{1 / n}=1$$ Hint: Define \(\alpha_{n}=n^{1 / n}-1\) and use the Binomial Formula to show that for each index \(n\) $$n=\left(1+\alpha_{n}\right)^{n} \geq 1+[n(n-1) / 2] \alpha_{n}^{2}$$

Examine the proof of the theorem that sequential compactness implies compactness and show that the only property of the sets \(I_{n}\) in the cover that we used was that if a point \(x\) lies in \(I_{n}\), then there is an open interval \(J\) centered at the point that also lies in \(I_{n}\). A set having this property is called open.

For \(c>0,\) consider the quadratic equation $$x^{2}-x-c=0, \quad x>0$$ Define the sequence \(\left\\{x_{n}\right\\}\) recursively by fixing \(x_{1}>0\) and then, if \(n\) is an index for which \(x_{n}\) has been defined, defining$$x_{n+1}=\sqrt{c+x_{n}}$$ Prove that the sequence \(\left\\{x_{n}\right\\}\) converges monotonically to the solution of the above equation.

Define the sequence \(\left\\{s_{n}\right\\}\) by $$s_{n}=\frac{1}{2 \cdot 1}+\frac{1}{3 \cdot 2}+\cdots+\frac{1}{(n+1)(n)} $$ for every index \(n\) Prove that $$\lim _{n \rightarrow \infty} s_{n}=1$$

Let \(\left\\{a_{n}\right\\}\) be a sequence of real numbers. Suppose that for each positive number \(c\) there is an index \(N\) such that $$a_{n}>c$$ for all indices \(n \geq N\) When this is so, the sequence \(\left\\{a_{n}\right\\}\) is said to converge to infinity, and we write $$\lim _{n \rightarrow \infty} a_{n}=\infty$$ Prove the following: a. \(\lim _{n \rightarrow \infty}\left[n^{3}-4 n^{2}-100 n\right]=\infty\) $$\text { b. } \lim _{n \rightarrow \infty}\left[\sqrt{n}-\frac{1}{n^{2}}+4\right]=\infty$$

Discuss the convergence to infinity of each of the following sequences: a. \(\\{\sqrt{n+1}-\sqrt{n}\\}\) b. \(\\{(\sqrt{n+1}-\sqrt{n}) \sqrt{n}\\}\) c. \(\\{(\sqrt{n+1}-\sqrt{n}) n\\}\)

For a sequence \(\left\\{a_{n}\right\\}\) of positive numbers show that $$\lim _{n \rightarrow \infty} a_{n}=\infty$$ if and only if \(\lim _{n \rightarrow \infty}\left[\frac{1}{a_{n}}\right]=0\)

(The Convergence of Cesaro Averages.) Suppose that the sequence \(\left\\{a_{n}\right\\}\) converges to \(a\). Define the sequence \(\left\\{\sigma_{n}\right\\}\) by $$\sigma_{n}=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}$$ for every index \(n\) Prove that the sequence \(\left\\{\sigma_{n}\right\\}\) also converges to \(a\).