Comments on Chapter 2

Page 41

The second sentence of the second paragraph is not grammatical. The following would be correct: “For example, as sequences, 1, 2, 3, 4, … and 2, 1, 3, 4, … are different, but as sets, {1, 2, 3, 4, …} and {2, 1, 3, 4, …} are identical.”

Page 43

In line 1, for “Remark 2.1” read “Example 2.2ii”.

The proof of Remark 2.4 illustrates a common device in the subject: you can apply Definition 2.1 with the arbitrary positive number \(\epsilon\) replaced by \(\epsilon/2\).

Page 46, Exercise 2.1.7

There is a typographical error in part b: the symbol \(n\) is in the wrong font.

Page 47, Example 2.10

In line 3 of the Solution, we know that \(2^n \gt n\) by Exercise 1.4.3b on page 28.

Page 56, proof of Theorem 2.26

In the line preceding equation (2), the inequality \(0 \le k\le m\) should really be \(1 \le k \le m\), for \(x_{n_0}\) is undefined.

Page 57, Exercise 2.3.3

For “or as” read “as”.

Page 60

In Exercise 2.4.0c, there are unmatched delimiters: the sequence \(\{1/(x_n+y_n)\}\) is missing the closing brace.

Page 61

In Exercise 2.4.3, the names of all the sequences should be inside braces.

Page 63

In Case 1 of the proof of Theorem 2.35, citing the Squeeze Theorem is overkill, for the conclusion follows immediately from the definition of an infinite limit. (Take \(N\) equal to any natural number larger than \(M\).)

Page 64

The paragraph at the top of the page is overkill. The display at the bottom of page 63 implies that \(|x_k-x|\lt\epsilon/2\) when \(k\ge N\), so the desired conclusion that \(x_n \to x\) holds by the definition of limit.

The statement of Theorem 2.37 is misleading. The statement has to be understood in the universe of the extended real numbers. Thus “largest value” means “largest extended real number”. Moreover, the use of the word “converges” is in conflict with Definition 2.14, according to which a sequence with limit \(\infty\) is said to diverge to \(\infty\).
A more explicit statement of the content of the theorem is the following: “If a sequence is not bounded above, then the \(\limsup\) is \(\infty\), and there is a subsequence having limit \(\infty\); if a sequence is bounded above, and it is not the case that the sequence has limit \(-\infty\), then the \(\limsup\) is the largest real number to which some subsequence converges; if the sequence has limit \(-\infty\), then the \(\limsup\) is \(-\infty\), and every subsequence has limit \(-\infty\).” (A parallel formulation holds for the limit infimum.)

The proof of Theorem 2.37 formally is incomplete, for all that has been proved explicitly is the displayed equation in the statement of the theorem, which says that every limit of a subsequence lies between the \(\liminf\) and the \(\limsup\). That the \(\liminf\) and the \(\limsup\) actually are attained as limits of subsequences is the content of Theorem 2.35, which ought to be referenced in the proof of Theorem 2.37.

The proof of Remark 2.38 is redundant, for the stated inequality is contained in formula (7) higher up on the page.

Page 66

There is a minor typographical error in Exercise 2.5.7: a space is missing in \(\sup x_k\).

Harold P. Boas