3.11: Operations on Convergent Sequences
This page is a draft and is under active development.
Sequences in \(E^{1}\) and \(C\) can be added and multiplied termwise; for example, adding \(\left\{x_{m}\right\}\) and \(\left\{y_{m}\right\},\) one obtains the sequence with general term \(x_{m}+y_{m}\). This leads to important theorems, valid also for \(E^{n}\)(* and other normed spaces). Theorem 1 below states, roughly, that the limit of the sum \(\left\{x_{m}+y_{m}\right\}\) equals the sum of lim \(x_{m}\) and lim \(y_{m}\) (if these exist), and similarly for products and quotients (when they are defined).
Let \(x_{m} \rightarrow q, y_{m} \rightarrow r,\) and \(a_{m} \rightarrow a\) in \(E^{1}\) or \(C\) (the complex field\() .\) Then
(i) \(x_{m} \pm y_{m} \rightarrow q \pm r\);
(ii) \(a_{m} x_{m} \rightarrow a q\);
(iii) \(\frac{x_{m}}{a_{m}} \rightarrow \frac{q}{a}\) if \(a \neq 0\) and for all \(m \geq 1, a_{m} \neq 0\).
This also holds if the \(x_{m}, y_{m}, q,\) and \(r\) are vectors in \(E^{n}\) ("or in another normed space\(),\) while the \(a_{m}\) and a are scalars for that space.
- Proof
-
(i) By formula (2) of §14, we must show that
\[(\forall \varepsilon>0)(\exists k)(\forall m>k) \quad\left|x_{m} \pm y_{m}-(q \pm r)\right|<\varepsilon.\]
Thus we fix an arbitrary \(\varepsilon>0\) and look for a suitable \(k .\) since \(x_{m} \rightarrow q\) and \(y_{m} \rightarrow r,\) there are \(k^{\prime}\) and \(k^{\prime \prime}\) such that
\[\left(\forall m>k^{\prime}\right) \quad\left|x_{m}-q\right|<\frac{\varepsilon}{2}\]
and
\[\left(\forall m>k^{\prime \prime}\right) \quad\left|y_{m}-r\right|<\frac{\varepsilon}{2}\]
(as \(\varepsilon\) is arbitrary, we may as well replace it by \(\frac{1}{2} \varepsilon ) .\) Then both inequalities hold for \(m>k, k=\max \left(k^{\prime}, k^{\prime \prime}\right) .\) Adding them, we obtain
\[(\forall m>k) \quad\left|x_{m}-q\right|+\left|y_{m}-r\right|<\varepsilon.\]
Hence by the triangle law,
\[\left|x_{m}-q \pm\left(y_{m}-r\right)\right|<\varepsilon, \text{ i.e., } \left|x_{m} \pm y_{m}-(q \pm r)\right|<\varepsilon \text{ for } m>k,\]
as required. \(\square\)
This proof of (i) applies to sequences of vectors as well, without any change.
The proof of (ii) and (iii) is sketched in Problems 1-4 below.
Note 1. By induction, parts (i) and (ii) hold for sums and products of any finite (but fixed) number of suitable convergent sequences.
Note 2. The theorem does not apply to infinite limits \(q, r, a\).
Note 3. The assumption \(a \neq 0\) in Theorem 1\((\) iii) is important. It ensures not only that \(q / a\) is defined but also that at most finitely many \(a_{m}\) can vanish (see Problem 3). Since we may safely drop a finite number of terms (see Note 2 in §14), we can achieve that no\(a_{m}\) is \(0,\) so that \(x_{m} / a_{m}\) is defined. It is with this understanding that part (iii) of the theorem has been formulated. The next two theorems are actually special cases of more general propositions to be proved in Chapter 4, §§3 and 5. Therefore, we only state them here, leaving the proofs as exercises, with some hints provided.
(componentwise convergence). We have \(\overline{x}_{m} \rightarrow \overline{p}\) in \(E^{n}\left(^{*} C^{n}\right)\) iff each of the \(n\) components of \(\overline{x}_{m}\) tends to the corresponding component of \(\overline{p}\), i.e., iff \(x_{m k} \rightarrow p_{k}, k=1,2, \ldots, n,\) in \(E^{1}(C) .\) (See Problem 8 for hints.)
Every monotone sequence \(\left\{x_{n}\right\} \subseteq E^{*}\) has a finite or infinite limit, which equals sup_ \(_{n} x_{n}\) if \(\left\{x_{n}\right\} \uparrow\) and inf \(_{n} x_{n}\) if \(\left\{x_{n}\right\} \downarrow .\) If \(\left\{x_{n}\right\}\) is monotone and bounded in \(E^{1},\) its limit is finite \((b y\) Corollary 1 of Chapter 2, §13).
The proof was requested in Problem 9 of Chapter 2, §13. See also Chapter 4, §5, Theorem 1. An important application is the following.
(the number e).
Let \(x_{n}=\left(1+\frac{1}{n}\right)^{n}\) in \(E^{1} .\) By the binomial theorem,
\[\begin{aligned} x_{n} &= 1+1+\frac{n(n-1)}{2 ! n^{2}}+\frac{n(n-1)(n-2)}{3 ! n^{3}}+\cdots + \frac{n(n-1) \cdots(n-(n-1))}{n ! n^{n}} \\[12pt] &= 2+\left(1-\frac{1}{n}\right) \frac{1}{2 !}+\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \frac{1}{3 !}+\cdots + \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right) \cdots\left(1-\frac{n-1}{n}\right) \frac{1}{n !} \end{aligned}\]
If \(n\) is replaced by \(n+1,\) all terms in this expansion increase, as does their number. Thus \(x_{n}<x_{n+1},\) i.e., \(\left\{x_{n}\right\} \uparrow .\) Moreover, for \(n>1\),
\[\begin{aligned} 2<x_{n}&<2+\frac{1}{2 !}+\cdots+\frac{1}{n !} \leq 2+\frac{1}{2}+\cdots+\frac{1}{2^{n-1}} \\[12pt] &= 2+\frac{1}{2}\left(1+\cdots+\frac{1}{2^{n-2}}\right)=2+\frac{1}{2} \frac{1-\left(\frac{1}{2}\right)^{n-1}}{\frac{1}{2}}<2+1=3 \end{aligned}\]
Thus \(2<x_{n}<3\) for \(n>1 .\) Hence \(2<\sup _{n} x_{n} \leq 3 ;\) and by Theorem \(3,\) \(\sup _{n} x_{n}=\lim x_{n} .\) This limit, denoted by \(e,\) plays an important role in analysis. It can be shown that it is irrational, and (to within \(10^{-20} )\) \(e=2.71828182845904523536 \ldots\) In any case,
\[2<e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n} \leq 3.\]
The following corollaries are left as exercises for the reader.
Suppose \(\lim x_{m}=p\) and \(\lim y_{m}=q\) exist in \(E^{*}\).
(a) If \(p>q,\) then \(x_{m}>y_{m}\) for all but finitely many \(m\).
(b) If \(x_{m} \leq y_{m}\) for infinitely many \(m,\) then \(p \leq q ;\) i.e., \(\lim x_{m} \leq \lim y_{m}\).
This is known as passage to the limit in inequalities. Caution: The strict inequalities \(x_{m}<y_{m}\) do not imply \(p<q\) but only \(p \leq q .\) For example, let
\[x_{m}=\frac{1}{m} \text{ and } y_{m}=0.\]
Then
\[(\forall m) \quad x_{m}>y_{m};\]
yet \(\lim x_{m}=\lim y_{m}=0.\)
Let \(x_{m} \rightarrow p\) in \(E^{*},\) and let \(c \in E^{*}\) (finite or not). Then the following are true:
(a) If \(p>c\) (respectively,\(p<c)\), we have \(x_{m}>c (x_{m}<c)\) for all but finitely many \(m\).
(b) If \(x_{m} \leq c\) (respectively, \(x_{m} \geq c\)) for infinitely many \(m,\) then \(p \leq c\) \((p \geq c)\).
One can prove this from Corollary \(1,\) with \(y_{m}=c\) (or \(x_{m}=c\)) for all \(m\).
(rule of intermediate sequence). If \(x_{m} \rightarrow p\) and \(y_{m} \rightarrow p\) in \(E^{*}\) and if \(x_{m} \leq z_{m} \leq y_{m}\) for all but finitely many \(m,\) then also \(z_{m} \rightarrow p\).
(continuity of the distance function). If
\[x_{m} \rightarrow p \text{ and } y_{m} \rightarrow q \text{ in a metric space } (S, \rho),\]
then
\[\rho\left(x_{m}, y_{m}\right) \rightarrow \rho(p, q) \text{ in } E^{1}.\]
- Proof
-
Hint: Show that
\[\left|\rho\left(x_{m}, y_{m}\right)-\rho(p, q)\right| \leq \rho\left(x_{m}, p\right)+\rho\left(q, y_{m}\right) \rightarrow 0\]
by Theorem 1.