Baby Rudin Problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 19 March 2014

Chapter 3

1. Prove that convergence of $\left\{s_n\right\}$ implies convergence of $\left\{\left|s_n\right|\right\}\text{.}$ Is the converse true?
2. Calculate $\underset{n\to \infty }{\text{lim}}(\sqrt{n^2+n}-n)\text{.}$
3. If $s_1=\sqrt{2},$ and $$s_{n+1}=\sqrt{2+\sqrt{s_n}}(n=1,2,3,\dots ),$$ prove that $\left\{s_n\right\}$ converges, and that $s_n<2$ for $n=1,2,3,\dots \text{.}$
4. Find the upper and lower limits of the sequence $\left\{s_n\right\}$ defined by $$s_1=0\text{;}{s}_{2m}=\frac{s_{2m-1}}{2}\text{;}{s}_{2m+1}=\frac{1}{2}+s_{2m}\text{.}$$
5. For any two real sequences $\left\{a_n\right\},\left\{b_n\right\},$ prove that $$\underset{n\to \infty }{\text{lim sup}}\hspace{0.17em}(a_n+b_n)\le \underset{n\to \infty }{\text{lim sup}}\hspace{0.17em}{a}_n+\underset{n\to \infty }{\text{lim sup}}\hspace{0.17em}{b}_n,$$ provided the sum on the right is not of the form $\infty -\infty \text{.}$
6. Investigate the behavior (convergence or divergence) of $\sum a_n$ if
   1. $a_n=\sqrt{n+1}-\sqrt{n}\text{;}$
   2. $a_n=\frac{\sqrt{n+1}-\sqrt{n}}{n}\text{;}$
   3. $a_n={(\sqrt[n]{n}-1)}^n\text{;}$
   4. $a_n=\frac{1}{1+z^n},$ for complex values of $z\text{.}$
7. Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n\ge 0\text{.}$
8. If $\sum a_n$ converges, and if $\left\{b_n\right\}$ is monotonic and bounded, prove that $\sum a_n{b}_n$ converges.
9. Find the radius of convergence of each of the following power series:
   1. $\sum n^3{z}^n,$
   2. $\sum \frac{2^n}{n!}{z}^n,$
   3. $\sum \frac{2^n}{n^2}{z}^n,$
   4. $\sum \frac{n^3}{3^n}{z}^n\text{.}$
10. Suppose that the coefficients of the power series $\sum a_n{z}^n$ are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most $1\text{.}$
11. Suppose $a_n>0,$ $s_n=a_1+\cdots +a_n,$ and $\sum a_n$ diverges.
    1. Prove that $\sum \frac{a_n}{1+a_n}$ diverges.
    2. Prove that $$\frac{a_{N+1}}{s_{N+1}}+\cdots +\frac{a_{N+k}}{s_{N+k}}\ge 1-\frac{s_N}{s_{N+k}}$$ and deduce that $\sum \frac{a_n}{s_n}$ diverges.
    3. Prove that $$\frac{a_n}{s_n^2}\le \frac{1}{s_{n-1}}-\frac{1}{s_n}$$ and deduce that $\sum \frac{a_n}{s_n^2}$ converges.
    4. What can be said about $$\sum \frac{a_n}{1+n{a}_n}\text{and}\sum \frac{a_n}{1+n^2{a}_n}\text{?}$$
12. Suppose $a_n>0$ and $\sum a_n$ converges. Put $$r_n=\sum _{m=n}^{\infty }{a}_m\text{.}$$
    1. Prove that $$\frac{a_m}{r_m}+\cdots +\frac{a_n}{r_n}>1-\frac{r_n}{r_m}$$ if $m<n,$ and deduce that $\sum \frac{a_n}{r_n}$ diverges.
    2. Prove that $$\frac{a_n}{\sqrt{r_n}}<2(\sqrt{r_n}-\sqrt{r_{n+1}})$$ and deduce that $\sum \frac{a_n}{\sqrt{r_n}}$ converges.
13. Prove that the Cauchy product of two absolutely convergent series converges absolutely.
14. If $\left\{s_n\right\}$ is a complex sequence, define its arithmetic means ${\sigma }_n$ by $${\sigma }_n=\frac{s_0+s_1+\cdots +s_n}{n+1}(n=0,1,2,\dots )\text{.}$$
    1. If $\text{lim}\hspace{0.17em}{s}_n=s,$ prove that $\text{lim}\hspace{0.17em}{\sigma }_n=s\text{.}$
    2. Construct a sequence $\left\{s_n\right\}$ which does not converge, although $\text{lim}\hspace{0.17em}{\sigma }_n=0\text{.}$
    3. Can it happen that $s_n>0$ for all $n$ and that $\text{lim sup}\hspace{0.17em}{s}_n=\infty ,$ although $\text{lim}\hspace{0.17em}{\sigma }_n=0\text{?}$
    4. Put $a_n=s_n-s_{n-1},$ for $n\ge 1\text{.}$ Show that $$s_n-{\sigma }_n=\frac{1}{n+1}\sum _{k=1}^{n}k{a}_k\text{.}$$ Assume that $\text{lim}\left(n{a}_n\right)=0$ and that $\left\{{\sigma }_n\right\}$ converges. Prove that $\left\{s_n\right\}$ converges. [This gives a converse of (a), but under the additional assumption that $n{a}_n\to 0\text{.]}$
    5. Define the last conclusion from a weaker hypothesis: Assume $M<\infty ,$ $\left|n{a}_n\right|\le M$ for all $n,$ and $\text{lim}\hspace{0.17em}{\sigma }_n=\sigma \text{.}$ Prove that $\text{lim}\hspace{0.17em}{s}_n=\sigma ,$ by completing the following outline: If $m<n,$ then $$s_n-{\sigma }_n=\frac{m+1}{n-m}({\sigma }_n-{\sigma }_m)+\frac{1}{n-m}\sum _{i=m+1}^n(s_n-s_i)\text{.}$$ For these $i,$ $$|s_n-s_i|\le \frac{(n-i)M}{i+1}\le \frac{(n-m-1)M}{m+2}\text{.}$$ Fix $\epsilon >0$ and associate with each $n$ the integer $m$ that satisfies $$m\le \frac{n-\epsilon }{1+\epsilon }<m+1\text{.}$$ Then $(m+1)/(n-m)\le 1/\epsilon$ and $|s_n-s_i|<M\epsilon \text{.}$ Hence $$\underset{n\to \infty }{\text{lim sup}}\hspace{0.17em}|s_n-\sigma |\le M\epsilon \text{.}$$ Since $\epsilon$ was arbitrary, $\text{lim}\hspace{0.17em}{s}_n=\sigma \text{.}$
15. Definition 3.21 can be extended to the case in which the $a_n$ lie in some fixed $R^k\text{.}$ Absolute convergence is defined as convergence of $\sum \left|{\text{a}}_n\right|\text{.}$ Show that Theorems 3.22, 3.23, 3.25(a), 3.33, 3.34, 3.42, 3.45, 3.47, and 3.55 are true in this more general setting. (Only slight modifications are required in any of the proofs.)
16. Fix a positive number $\alpha \text{.}$ Choose $x_1>\sqrt{\alpha },$ and define $x_2,x_3,x_4,\dots ,$ by the recursion formula $$x_{n+1}=\frac{1}{2}(x_n+\frac{\alpha }{x_n})\text{.}$$
    1. Prove that $\left\{x_n\right\}$ decreases monotonically and that $\text{lim}\hspace{0.17em}{x}_n=\sqrt{\alpha }\text{.}$
    2. Put ${\epsilon }_n=x_n-\sqrt{\alpha },$ and show that $${\epsilon }_{n+1}=\frac{{\epsilon }_n^2}{2x_n}<\frac{{\epsilon }_n^2}{2\sqrt{\alpha }}$$ so that, setting $\beta =2\sqrt{\alpha },$ $${\epsilon }_{n+1}<\beta {\left(\frac{{\epsilon }_1}{\beta }\right)}^{2^n}(n=1,2,3,\dots )\text{.}$$
    3. This is a good algorithm for computing square roots, since the recursion formula is simple and the convergence is extremely rapid. For example, if $\alpha =3$ and $x_1=2,$ show that ${\epsilon }_1/\beta <\frac{1}{10}$ and that therefore $${\epsilon }_5<4·{10}^{-16},{\epsilon }_6<4·{10}^{-32}\text{.}$$
17. Fix $\alpha >1\text{.}$ Take $x_1>\sqrt{\alpha },$ and define $$x_{n+1}=\frac{\alpha +x_n}{1+x_n}=x_n+\frac{\alpha -x_n^2}{1+x_n}\text{.}$$
    1. Prove that $x_1>x_3>x_5>\cdots \text{.}$
    2. Prove that $x_2<x_4<x_6<\cdots \text{.}$
    3. Prove that $\text{lim}\hspace{0.17em}{x}_n=\sqrt{\alpha }\text{.}$
    4. Compare the rapidity of convergence of this process with the one described in Exercise 16.
18. Replace the recursion formula of Exercise 16 by $$x_{n+1}=\frac{p-1}{p}{x}_n+\frac{\alpha }{p}{x}_n^{-p+1}$$ where $p$ is a fixed positive integer, and describe the behavior of the resulting sequences $\left\{x_n\right\}\text{.}$
19. Associate to each sequence $a=\left\{{\alpha }_n\right\},$ in which ${\alpha }_n$ is $0$ or $2,$ the real number $$x\left(a\right)=\sum _{n=1}^{\infty }\frac{{\alpha }_n}{3^n}\text{.}$$ Prove that the set of all $x\left(a\right)$ is precisely the Cantor set described in Sec. 2.44.
20. Suppose $\left\{p_n\right\}$ is a Cauchy sequence in a metric space $X,$ and some subsequence $\left\{p_{n_i}\right\}$ converges to a point $p\in X\text{.}$ Prove that the full sequence $\left\{p_n\right\}$ converges to $p\text{.}$
21. Prove the following analogue of Theorem 3.10(b): If $\left\{E_n\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X,$ if $E_n\supset E_{n+1},$ and if $$\underset{n\to \infty }{\text{lim}}\hspace{0.17em}\text{diam}\hspace{0.17em}{E}_n=0,$$ then $\bigcap _1^{\infty }{E}_n$ consists of exactly one point.
22. Suppose $X$ is a nonempty complete metric space, and $\left\{G_n\right\}$ is a sequence of dense open subsets of $X\text{.}$ Prove Baire's theorem, namely, that $\bigcap _1^{\infty }{G}_n$ is not empty. (In fact, it is dense in $X\text{.)}$ Hint: Find a shrinking sequence of neighborhoods $E_n$ such that ${\bar{E}}_n\subset G_n,$ and apply Exercise 21.
23. Suppose $\left\{p_n\right\}$ and $\left\{q_n\right\}$ are Cauchy sequences in a metric space $X\text{.}$ Show that the sequence $\left\{d(p_n,q_n)\right\}$ converges. Hint: For any $m,n,$ $$d(p_n,q_n)\le d(p_n,p_m)+d(p_m,q_m)+d(q_m,q_n)\text{;}$$ it follows that $$\mid d(p_n,q_n)-d(p_m,q_m)\mid$$ is small if $m$ and $n$ are large.
24. Let $X$ be a metric space.
    1. Call two Cauchy sequences $\left\{p_n\right\},\left\{q_n\right\}$ in $X$ equivalent if $$\underset{n\to \infty }{\text{lim}}\hspace{0.17em}d(p_n,q_n)=0\text{.}$$ Prove that this is an equivalence relation.
    2. Let $X^{*}$ be the set of all equivalence classes so obtained. If $P\in X^{*},$ $Q\in X^{*},$ $\left\{p_n\right\}\in P,$ $\left\{q_n\right\}\in Q,$ define $$\Delta (P,Q)=\underset{n\to \infty }{\text{lim}}d(p_n,q_n)\text{;}$$ by Exercise 23, this limit exists. Show that the number $\Delta (P,Q)$ is unchanged if $\left\{p_n\right\}$ and $\left\{q_n\right\}$ are replaced by equivalent sequences, and hence that $\Delta$ is a distance function in $X^{*}\text{.}$
    3. Prove that the resulting metric space $X^{*}$ is complete.
    4. For each $p\in X,$ there is a Cauchy sequence all of whose terms are $p\text{;}$ let $P_p$ be the element of $X^{*}$ which contains this sequence. Prove that $$\Delta (P_p,P_q)=d(p,q)$$ for all $p,q\in X\text{.}$ In other words, the mapping $\phi$ defined by $\phi \left(p\right)=P_p$ is an isometry (i.e., a distance-preserving mapping) of $X$ into $X^{*}\text{.}$
    5. Prove that $\phi \left(X\right)$ is dense in $X^{*},$ and that $\phi \left(X\right)=X^{*}$ if $X$ is complete. By (d), we may identify $X$ and $\phi \left(X\right)$ and thus regard $X$ as embedded in the complete metric space $X^{*}\text{.}$ We call $X^{*}$ the completion of $X\text{.}$
25. Let $X$ be the metric space whose points are the rational numbers, with the metric $d(x,y)=|x-y|\text{.}$ What is the completion of this space? (Compare Exercise 24.)

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