Section 7.1 Sequences and Completeness
As in the real case, a
(complex) sequence is a function from the positive (sometimes the nonnegative) integers to the complex numbers. Its values are usually written as
\(a_n\) (as opposed to
\(a(n)\)) and we commonly denote the sequence by
\(\left( a_n
\right)_{n=1}^\infty\text{,}\) \(\left( a_n \right)_{n \geq 1}\text{,}\) or simply
\((a_n)\text{.}\) Considering such a sequence as a function of
\(n\text{,}\) the notion of convergence is merely a repeat of the definition we gave in
Section 3.2, adjusted to the fact that
\(n\) is an integer.
Definition 7.1.1.
Suppose \(\left( a_n \right)\) is a sequence and \(L \in
\C\) such that for all \(\epsilon>0\) there is an integer \(N\) such that for all \(n \geq N\text{,}\) we have \(\left| a_n - L \right| \lt \epsilon\text{.}\) Then the sequence \(\left( a_n \right)\) is convergent and \(L\) is its limit; in symbols we write
\begin{equation*}
\lim_{n \to \infty} a_n \ = \ L \,\text{.}
\end{equation*}
If no such
\(L\) exists then the sequence
\(\left( a_n
\right)\) is
divergent.
As in our previous definitions of limit, the limit of a sequence is unique if it exists. See
Exercise 7.5.7.
Example 7.1.2.
We claim that \(\ds \lim_{n \to \infty} \tfrac {i^n} n =
0\text{:}\) Given \(\epsilon > 0\text{,}\) choose \(N > \frac 1 \epsilon\text{.}\) Then for any \(n \geq N\text{,}\)
\begin{equation*}
\left| \frac{i^n} n - 0 \right| \ = \ \left| \frac{i^n} n
\right| \ = \ \frac {|i|^n} n \ = \ \frac 1 n \ \leq \ \frac 1 N \
\lt \ \epsilon \, \text{.}
\end{equation*}
To prove that a sequence
\((a_n)\) is divergent, we have to show the negation of the statement that defines convergence, that is: given any
\(L \in \C\text{,}\) there exists
\(\epsilon > 0\) such that, given any integer
\(N\text{,}\) there exists an integer
\(n\) such that
\(|a_n - L| \ge \epsilon\text{.}\) (If you have not negated many mathematical statements, this is worth meditating about.)
Example 7.1.3.
The sequence \(\left( a_n = i^n \right)\) diverges: Given \(L \in \C\text{,}\) choose \(\epsilon = \frac 1 2\text{.}\) We consider two cases: If \(\Re(L) \geq 0\text{,}\) then for any \(N\text{,}\) choose \(n \geq N\) such that \(a_n = -1\text{.}\) (This is always possible since \(a_{4k+2} = i^{4k+2} = -1\) for any \(k \geq 0\text{.}\)) Then
\begin{equation*}
\left| a_n - L \right| \ = \ \left| 1 + L \right| \ \geq \ 1 \
> \ \frac 1 2 \, \text{.}
\end{equation*}
If \(\Re(L) \lt 0\text{,}\) then for any \(N\text{,}\) choose \(n \geq N\) such that \(a_n = 1\text{.}\) (This is always possible since \(a_{4k} = i^{4k} = 1\) for any \(k > 0\text{.}\)) Then
\begin{equation*}
\left| a_n - L \right| \ = \ \left| 1 - L \right| \ > \ 1 \ > \
\frac 1 2 \, \text{.}
\end{equation*}
This proves that
\(\left( a_n = i^n \right)\) diverges.
Proposition 7.1.4.
Let \(\left( a_n \right)\) and \(\left( b_n \right)\) be convergent sequences and \(c \in \C\text{.}\) Then
-
\(\displaystyle \displaystyle \lim_{ n \to \infty } a_n + c \lim_{ n
\to \infty } b_n \ = \ \lim_{ n \to \infty } \left( a_n + c
\, b_n \right)\)
-
\(\displaystyle \displaystyle \lim_{ n \to \infty } a_n \cdot \lim_{
n \to \infty } b_n \ = \ \lim_{ n \to \infty } \left( a_n
\cdot b_n \right)\)
-
\(\displaystyle \frac{ \lim_{ n \to \infty } a_n }{
\lim_{ n \to \infty } b_n } \ = \ \lim_{ n \to \infty }
\left( \frac{ a_n }{ b_n } \right)\) provided
\(\lim_{ n \to \infty } b_n \neq 0\text{.}\)
-
\(\displaystyle \lim_{ n \to \infty } a_n \ = \ \lim_{ n
\to \infty } a_{ n+1 }\text{.}\)
Furthermore, if \(f: G \to \C\) is continuous at \(L := \lim_{ n \to \infty } a_n\) and all \(a_n \in
G\text{,}\) then
\begin{equation*}
\lim_{n\to\infty} f(a_n) \ = \ f(L) \,\text{.}
\end{equation*}
Again, the proof of this proposition is essentially a repeat from arguments we have given in
Chapter 2 and
Chapter 3, as you should convince yourself in
Exercise 7.5.4.
We will assume, as an axiom, that
\(\R\) is
complete. To phrase this precisely, we need the following.
Definition 7.1.5.
The sequence
\((a_n)\) is
monotone if it is either nondecreasing (
\(a_{n+1}\ge a_n\) for all
\(n\)) or nonincreasing (
\(a_{n+1}\le a_n\) for all
\(n\)).
There are many equivalent ways of formulating the completeness property for the reals. Here is what we’ll go by:
Axiom 7.1.6. Monotone Sequence Property.
Any bounded monotone sequence converges.
This axiom (or one of its many equivalent statements) gives arguably the most important property of the real number system; namely, that we can, in many cases, determine that a given sequence converges
without knowing the value of the limit. In this sense we can use the sequence to define a real number.
Example 7.1.7.
Consider the sequence \((a_n)\) defined by
\begin{equation*}
a_n \ := \ 1 + \frac 1 2 + \frac 1 6 + \dots + \frac 1 {n!} \,\text{.}
\end{equation*}
This sequence is increasing (by definition) and each
\(a_n \le 3\) by
Exercise 7.5.9. By the Monotone Sequence Property,
\((a_n)\) converges, which allows us to define one of the most famous numbers in all of mathematics,
\begin{equation*}
e \ := \ 1 + \lim_{ n \to \infty } a_n \,\text{.}
\end{equation*}
Example 7.1.8.
Fix
\(0\le r\lt 1\text{.}\) We claim that
\(\lim_{n\to\infty}r^n=0\text{:}\) First, the sequence
\((a_n = r^n)\) converges because it is decreasing and bounded below by
\(0\text{.}\) Let
\(L := \lim_{n\to\infty}r^n\text{.}\) By
Proposition 7.1.4,
\begin{equation*}
L \ = \ \lim_{n\to\infty}r^n \ = \ \lim_{n\to\infty} r^{n+1} \ = \ r
\lim_{n\to\infty}r^n \ = \ r \, L \, \text{.}
\end{equation*}
Thus
\((1-r)L=0\text{,}\) and so (since
\(1-r \ne 0\)) we conclude that
\(L=0\text{.}\)
We remark that the Monotone Sequence Property implies the
Least Upper Bound Property: every nonempty set of real numbers with an upper bound has a
least upper bound. The Least Upper Bound Property, in turn, implies the following theorem, which is often listed as a separate axiom.
Theorem 7.1.9. Archimedean Property.
If
\(x\) is any real number then there is an integer
\(N\) that is greater than
\(x\text{.}\)
For a proof, see
Exercise 7.5.10.
Theorem 7.1.9 essentially says that infinity is not part of the real numbers. Note that we already used
Theorem 7.1.9 in
Example 7.1.2. The Archimedean Property underlies the construction of an infinite decimal expansion for any real number, while the Monotone Sequence Property shows that any such infinite decimal expansion actually converges to a real number.
We close this discussion of limits with a pair of standard limits. The first of these can be established by calculus methods (such as L’Hôpital’s rule (
Theorem A.0.11), by treating
\(n\) as the variable); both of them can be proved by more elementary considerations. Either way, we leave the proof of the following to
Exercise 7.5.11.
Proposition 7.1.10.
-
Exponentials beat polynomials: for any polynomial \(p(n)\) (with complex coefficients) and any \(c \in \C\) with \(\abs{c} > 1\text{,}\)
\begin{equation*}
\lim_{n\to\infty}\frac{p(n)}{c^n}\ = \ 0 \,\text{.}
\end{equation*}
-
Factorials beat exponentials: for any \(c \in \C\text{,}\)
\begin{equation*}
\lim_{n\to\infty} \frac{c^n}{n!}\ = \ 0 \,\text{.}
\end{equation*}
