The Theory of Convergence

2.1 The definition of the limit of a sequence[1]

[1]A definition equivalent to this was first given by John Wallis in 1655. [Opera, I (1695). p. 382.] ↩

Let \(z_1\), \(z_2\), \(z_3, \dots\) be an unending sequence of numbers, real or complex. Then, if a number \(l\) exists such that, corresponding to every positive[2] number \(\epsilon\), no matter how small, a number \(n_0\) can be found, such that \[ \left|\, z_n-l \,\right| < \epsilon \] for all values of \(n\) greater than \(n_0\), the sequence \((z_n)\) is said to tend to the limit \(l\) as \(n\) tends to infinity.

[2]The number zero is excluded from the class of positive numbers. ↩

Symbolic forms of the statement[3] ‘the limit of the sequence \((z_n)\), as \(n\) tends to infinity, is \(l\)’ are: \[ \lim_{n \rightarrow \infty} z_n= l, \quad \lim z_n = l, \quad z_n \rightarrow l \,\text{ as }\, n \rightarrow \infty . \]

[3]The arrow notation is due to Leathem, Camb. Math. Tracts, No. 1. ↩

If the sequence be such that, given an arbitrary number \(N\) (no matter how large), we can find \(n_0\) such that \(\left|\,z_n\,\right| > N\) for all values of \(n\) greater than \(n_0\), we say that ‘\(\left|\, z_n \,\right|\) tends to infinity as \(n\) tends to infinity’, and we write \[ \left|\, z_n \,\right| \rightarrow \infty .\]

In the corresponding case when \(-x_n>N\) when \(n> n_0\) we say that \(x_n \rightarrow -\infty\).

If a sequence of real numbers does not tend to a limit or to \(\infty\) or to \(-\infty\), the sequence is said to oscillate.

2.11 Definition of the phrase ‘of the order of’

If \((\zeta_n)\) and \((z_n)\) are two sequences such that a number \(n_0\) exists such that \(\left|\,(\zeta_n \left/z_n \right.)\,\right| < K\) whenever \(n > n_0\), where \(K\) is independent of \(n\), we say that \(\zeta_n\) is ‘of the order of’ \(z_n\), and we write[4] \[ \zeta_n = O(z_n); \] thus \[ \frac{15n +19}{1 + n^3} = O \left (\frac{1}{n^2} \right ). \] If \(\lim(\zeta_n\left/z_n \right.) = 0\), we write \(\zeta_n = o(z_n)\).

[4]This notation is due to Bachmann, Zahlentheorie (1894), p. 401, and Landau, Primzahlen, I. (1909), p. 59. ↩

2.2 The limit of an increasing sequence

Let \((x_n)\) be a sequence of real numbers such that \(x_{n+1} \geq x_n\) for all values of \(n\); then the sequence tends to a limit or else tends to infinity (and so it does not oscillate).

Let \(x\) be any rational-real number; then either:

  1. \(x_n \geq x\) for all values of \(n\) greater than some number \(n_0\) depending on the value of \(x\).
  2. \(x_n < x\) for every value of \(n\).

If (ii) is not the case for any value of \(x\) (no matter how large), then \(x_n \rightarrow \infty\).

But if values of \(x\) exist for which (ii) holds, we can divide the rational numbers into two classes, the \(L\)-class consisting of those rational numbers \(x\) for which (i) holds and the \(R\)-class of those rational numbers \(x\) for which (ii) holds. This section defines a real number \(\alpha\), rational or irrational.

And if \(\epsilon\) be an arbitrary positive number, \(\alpha - \frac{1}{2}\epsilon\) belongs to the \(L\)-class which defines \(\alpha\), and so we can find \(n_1\) such that \(x_n \geq \alpha - \frac{1}{2}\epsilon\) whenever \(n > n_1\); and \(\alpha + \frac{1}{2}\epsilon\) is a member of the \(R\)-class and so \(x_n \leq \alpha + \frac{1}{2}\epsilon\). Therefore, whenever \(n > n_1\), \( \left|\, \alpha - x_n \,\right| < \epsilon .\) Therefore \(x_n \rightarrow \alpha \).[5]

[5]Editor’s Note: Since Whittaker and Watson define the \(L\)-class and \(R\)-class as sets of rational numbers, we cannot ignore the possibility that \(\alpha - \frac{1}{2}\epsilon\) is irrational. Nonetheless, since \( \alpha> \alpha- \frac{1}{2}\epsilon\), there are rationals in the \(L\)-class of \(\alpha\) that are larger than \(\alpha - \frac{1}{2}\epsilon\), say, \(r = \alpha-\epsilon_1 \). Since \( \epsilon_1<\frac{1}{2}\epsilon\), using \( \epsilon_1 \) in place of \( \frac{1}{2}\epsilon\) will let this argument work. ↩

Corollary. A decreasing sequence tends to a limit or to \(-\infty\)

Example 1. If \(\lim z_n = l, \; \lim z'_n = l'\), then \(\lim\:\!(z_n + z'_n) = l + l'\). For, given \(\epsilon\), we can find \(n\) and \(n'\) such that
  1. when \(m > n, \; \left|\, z_m - l \,\right| < \frac{1}{2} \epsilon ,\)
  2. when \(m > n', \; \left|\, z'_m- l' \,\right| < \frac{1}{2} \epsilon .\)
Let \(n_1\) be the greater of \(n\) and \(n'\); then, when \(m>n_1\),

\[ \left|\, (z_m + z'_m) - (l + l') \,\right| \leq \left|\, (z_n - l) \,\right| + \left|\, (z'_m - l') \,\right| < \epsilon ; \] and this is the condition that \(\lim \:\!(z_n + z'_m)= l + l'\).

Example 2. Prove similarly that
  1. \(\lim \:\!(z_n - z'_n) = l - l',\)
  2. \( \lim \:\!(z_n z'_n) = l \, l',\)
  3. and, if \( l' \neq 0 , \; \lim \:\!(z_n \left/ z'_n \right.) = \left. l \right/ l'.\)
Example 3. If \( 0 < x < 1\), \(x^n \rightarrow 0\). For if \(x = (1+a)^{-1}\), \(a > 0\) and \[ 0 < x^n =\frac{1}{(1+a)^n} <\frac{1}{1+na}, \] by the binomial theorem for a positive integral index. And it is obvious that, given a positive number \(\epsilon\), we can choose \(n_0\) such that \((1 + na)^{-1} < \epsilon \) when \(n > n_0\); and so \(x^n \rightarrow 0\).

2.21 Limit-points and the Bolzano-Weierstrass theorem[6]

[6]This theorem, frequently ascribed to Weierstrass, was proved by Bolzano, Abh. der k. böhmischen Gen. der Wiss. v. (1817). [Reprinted in Klassiker der Exakten Wiss., No. 153.] It seems to have been known to Cauchy. ↩

Let \((x_n)\) be a sequence of real numbers. if any number \(G\) exists such that, for every positive value of \(\epsilon\), no matter how small, an unlimited number of terms of the sequence can be found such that \[ G - \epsilon < x_n < G + \epsilon, \] then \(G\) is called a limit-point, or cluster-point, of the sequence.

Bolzano’s theorem is that, if \(\lambda \leq x_n \leq \rho\), where \(\lambda\), \(\rho\) are independent of \(n\), then the sequence \((x_n)\) has at least one limit-point.

To prove the theorem, choose a section in which (i) the \(R\)-class consists of all the rational numbers which are such that, if \(A\) be any one of them, there are only a limited number of terms \(x_n\) satisfying \(x_n >A\); and (ii) the \(L\)-class is such that there are an unlimited number of terms \(x_n\) such that \(x_n \geq a\) for all members \(a\) of the \(L\)-class.

This section defines a real number \(G\); and, if \(\epsilon\) be an arbitrary positive number, \(G - \frac{1}{2} \epsilon\) and \(G + \frac{1}{2} \epsilon\) are members of the \(L\) and \(R\) classes respectively,[7] and so there are an unlimited number of terms of the sequence satisfying \[ \textstyle G - \epsilon < G - \frac{1}{2} \epsilon \leq x_n \leq G + \frac{1}{2} \epsilon< G + \epsilon, \] and so \(G\) satisfies the condition that it should be a limit-point.

[7]Editor’s Note: Just as earlier, a small adjustment needs to be made in the event \(G - \frac{1}{2} \epsilon\) and \(G + \frac{1}{2} \epsilon\) are irrational. ↩
2.211 Definition of ‘the greatest of the limits’

The number \(G\) obtained in §2.21 is called ‘the greatest of the limits of the sequence \((x_n)\)’. The sequence \((x_n)\) cannot have a limit-point greater than \(G\); for if \(G' \) were such a limit-point, and \(\epsilon =\frac{1}{2}(G -G' )\), \(G' —\epsilon\) is a member of the \(R\)-class defining \(G\), so that there are only a limited number of terms of the sequence which satisfy \(x_n > G' -\epsilon\). This condition is inconsistent with \(G' \) being a limit-point. We write \[ G= \varlimsup_{n \rightarrow \infty} x_n. \] The ‘least of the limits’, \(L\), of the sequence (written \(\displaystyle \varliminf_{n \rightarrow \infty} x_n\) ) is defined to be \[ - \varlimsup_{n \rightarrow \infty} (-x_n). \]

2.22 Cauchy’s theorem on the necessary and sufficient condition for the existence of a limit.[8]

[8]Analyse Algébrique (1821), p. 125. ↩

We shall now shew that the necessary and sufficient condition for the existence of a limiting value of a sequence of numbers \(z_1\), \(z_2\), \(z_3, \dots\) is that, corresponding to any given positive number \(\epsilon\), however small, it shall be possible to find a number \(n\) such that \[ \left|\, z_{n+p}-z_n \,\right| < \epsilon \] for all positive integral values of \(p\). This result is one of the most important and fundamental theorems of analysis. It is sometimes called the Principle of Convergence.

First, we have to shew that this condition is necessary, i.e. that it is satisfied whenever a limit exists. Suppose then that a limit \(l\) exists; then (§2.1) corresponding to any positive number \(\epsilon\), however small, an integer \(n\) can be chosen such that \[ \textstyle \left|\, z_n - l \,\right| < \frac{1}{2} \epsilon, \quad \left|\, z_{n+p} - l \,\right| < \frac{1}{2} \epsilon , \] for all positive values of \(p\); therefore \[ \begin{align*} \left|\, z_{n+p} - z_n \,\right| &= \left|\, (z_{n+p} - l) - (z_n - l) \,\right| \\ & \leq \left|\, z_{n+p} - l \,\right| + \left|\, z_n - l \,\right| < \epsilon , \\ \end{align*} \] which shews the necessity of the condition \[ \left|\, z_{n+p} - z_n \,\right| < \epsilon , \] and thus establishes the first half of the theorem.

Secondly, we have to prove[9] that this condition is sufficient, i.e. that if it is satisfied, then a limit exists.

[9]This proof is given by Stolz and Gmeiner, Theoretische Arithmetik, ii. (1902), p. 144. ↩

(I) Suppose that the sequence of real numbers \((x_n)\) satisfies Cauchy’s condition; that is to say that, corresponding to any positive number \(\epsilon\), an integer \(n\) can be chosen such that \[ \left|\, z_{n+p} - z_n \,\right| < \epsilon \] for all positive integral values of \(p\).

Let the value of \(n\), corresponding to the \(\epsilon\) equaling \(1\), be \(m\).

Let \(\lambda_1\), \(\rho_1\) be the least and greatest of \(x_1\), \(x_2,\, \dots , x_m\); then \[ \lambda_1 -1 < x_n < \rho_1 +1 , \] for all values of \(n\); write \(\lambda_1 -1=\lambda\), \( \rho_1 +1=\rho \).

Then, for all values of \(n\), \(\lambda < x_n < \rho\). Therefore by the theorem of §2.21, the sequence \((x_n)\) has at least one limit-point \(G\).

Further, there cannot be more than one limit-point; for if there were two, \(G\) and \(H \) \( (H < G)\), take \(\epsilon < \frac{1}{4}(G - H)\). Then, by hypothesis, a number \(n\) exists such that \(\left|\, x_{n+p} - x_n \,\right| < \epsilon\) for every positive value of \(p\). But since \(G\) and \(H\) are limit-points, positive numbers \(q\) and \(r\) exist such that \[ \left|\, G - x_{n+q} \,\right| < \epsilon, \quad \left|\, H - x_{n+r} \,\right| < \epsilon . \]

Then \( \left|\, G - x_{n+q} \,\right| + \left|\, x_{n+q} - x_n \,\right| + \left|\, x_n - x_{n+r} \,\right| + \left|\, H - x_{n+r} \,\right| < 4 \epsilon \). But, by §1.4, the sum on the left is greater than or equal to \(\left|\, G - H \,\right|\).

Therefore \(G - H < 4\epsilon\), which is contrary to hypothesis; so there is only one limit-point. Hence there are only a finite number of terms of the sequence outside the interval \((G - \delta, G + \delta)\), where \(\delta\) is an arbitrary positive number; for, if there were an unlimited number of such terms, these would have a limit-point which would be a limit-point of the given sequence and which would not coincide with \(G\); and therefore \(G\) is the limit of \((x_n)\).

(II) Now let the sequence \((z_n)\) of real or complex numbers satisfy Cauchy’s condition; and let \(z_n = x_n - i y_n\), where \(x_n\) and \(y_n\) are real; then for all values of \(n\) and \(p\) \[ \left|\, x_{n+p} - x_n \,\right| \leq \left|\, z_{n+p} - z_n \,\right|, \quad \left|\, y_{n+p} - y_n \,\right| \leq \left|\, z_{n+p} - z_n \,\right|. \]

Therefore the sequences of real numbers \((x_n)\) and \((y_n)\) satisfy Cauchy’s condition; and so, by (I), the limits of \((x_n)\) and \((y_n)\) exist. Therefore, by §2.2, example 1, the limit of \((z_n)\) exists. The result is therefore established.