Real Numbers and Limits of Sequences
1.1 Number Systems
The Peano axioms describe the arithmetic properties of the set of natural numbers \(\mathbb{N}\). The formal definition is as follows:
There exists an element \(0 \in \mathbb{N}\).
There exists a function \[ S : \mathbb{N} \to \mathbb{N} \] called the successor function, such that for every \(n \in \mathbb{N}\), \(S(n)\) represents the next natural number after \(n\).
Zero is not the successor of any number \[ \forall n \in \mathbb{N},\; S(n) \neq 0 \]
Successor function is injective (one-to-one) \[ \forall m, n \in \mathbb{N},\; S(m) = S(n) \implies m = n \]
Principle of mathematical induction. For any property \(P(n)\) defined on \(\mathbb{N}\): If \[ P(0) \text{ is true, and } \forall n \in \mathbb{N},\; P(n) \Rightarrow P(S(n)), \]
then \[ \forall n \in \mathbb{N},\; P(n) \text{ is true.} \]
Addition:
The addition \(+ : \mathbb{N} \times \mathbb{N} \to \mathbb{N}\) is defined recursively as follows:
\[ a + 0 = a \]
\[ a + S(b) = S(a + b) \]
Multiplication:
The multiplication \(\times : \mathbb{N} \times \mathbb{N} \to \mathbb{N}\) is defined recursively as follows:
\[ a \times 0 = 0 \]
\[ a \times S(b) = a \times b + a \]
Order Relation:
For all \(m, n \in \mathbb{N}\):
\[ m < S(n) \iff (m < n) \text{ or } (m = n). \]
This defines \(<\) recursively based on the successor structure.
Theorems:
\(a + (b + c) = (a + b) + c\).
\(a + 0 = 0 + a = a\).
\(a + b = b + a\).
\((a + b)\times c = a \times c + b \times c\).
\(a \times 1 = 1 \times a = a\).
\(a \times b = b \times a\).
\(a \times (b \times c) = (a \times b) \times c\).
Integers can be defined from natural numbers as equivalence classes of ordered pairs of natural numbers:
\[ \mathbb{Z} = \{ (a, b) \mid a, b \in \mathbb{N} \} \] We define an equivalence relation:
\[ (a, b) \sim (c, d) \iff a + d = b + c \]
- Each equivalence class \([(a, b)]\) represents the difference between \(a\) and \(b\). Thus we have: \[ \begin{aligned} 0 &= [(0, 0)], \\ 1 &= [(1, 0)], \\ -1 &= [(0, 1)], \\ a - b &= [(a, b)]. \end{aligned} \]
Addition:
\[ [(a, b)] + [(c, d)] = [(a + c,\, b + d)] \]
Multiplication:
\[ [(a, b)] \times [(c, d)] = [(ac + bd,\, ad + bc)] \]
These definitions are well-defined, meaning the result does not depend on the choice of representatives of the equivalence classes.
Order Relation:
\[ [(a,b)] < [(c,d)] \iff a + d < b + c. \]
Theorems:
\(a + b = b + a, \qquad a \times b = b \times a\).
\(a + (b + c) = (a + b) + c, \qquad a \times (b \times c) = (a \times b) \times c\).
\(a + 0 = a, \qquad a \times 1 = a\).
\(\forall a \in \mathbb{Z}, \; \exists (-a) \in \mathbb{Z} \text{ such that } a + (-a) = 0\).
\(a \times (b + c) = a \times b + a \times c\).
The set of rational numbers \(\mathbb{Q}\) is constructed from the integers \(\mathbb{Z}\) as the set of equivalence classes of ordered pairs of integers:
\[ \mathbb{Q} = \{ (a, b) \mid a \in \mathbb{Z},\; b \in \mathbb{Z} \setminus \{0\} \} \]
Each pair \((a,b)\) intuitively represents the fraction \(\dfrac{a}{b}\).
We define an equivalence relation: \[ (a,b) \sim (c,d) \iff ad = bc \]
Each equivalence class \([(a,b)]\) corresponds to a rational number. We write:
\[ \dfrac{a}{b} = [(a,b)], \quad \text{where } b \neq 0 \]
Addition:
\[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd} \]
Multiplication:
\[ \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd} \]
Additive Inverse:
\[ -\dfrac{a}{b} = \dfrac{-a}{b} \]
Multiplicative Inverse:
\[ \left(\dfrac{a}{b}\right)^{-1} = \dfrac{b}{a}, \quad a \neq 0 \]
Order relation:
To compare two rational numbers, let
\[ \dfrac{a}{b}, \; \dfrac{c}{d} \in \mathbb{Q}, \quad b > 0, \; d > 0. \]
We define the order relation \(<\) on \(\mathbb{Q}\) by:
\[ \dfrac{a}{b} < \dfrac{c}{d} \iff ad < bc. \]
Theorems:
\(a+b = b+a, \qquad a\times b = b\times a\).
\(a+(b+c) = (a+b)+c, \qquad a\times(b\times c) = (a\times b)\times c\).
\(a+0 = a, \qquad a\times1 = a\).
\(\forall a \in \mathbb{Q},\; \exists (-a) \text{ such that } a+(-a)=0\).
\(\forall a \in \mathbb{Q}\setminus\{0\},\; \exists a^{-1} \text{ such that } a\times a^{-1}=1\).
\(a\times(b+c) = a\times b + a\times c\).
A Dedekind cut in \(\mathbb{Q}\) is a partition \((A, B)\) of \(\mathbb{Q}\) satisfying:
\[ \begin{aligned} (1)\quad & A, B \subset \mathbb{Q}, \; A \cup B = \mathbb{Q}, \; A \cap B = \varnothing, \\[4pt] (2)\quad & \forall a \in A,\; \forall b \in B,\; a < b, \\[4pt] (3)\quad & A \text{ has no greatest element.} \end{aligned} \] Each real number is identified with one such cut \(A\), which intuitively represents “all rationals less than that real number.”
The set of all Dedekind cuts is denoted by:
\[ \mathbb{R} = \{ A \subset \mathbb{Q} \mid A \text{ is a Dedekind cut} \}. \]
- For example:
\[ \sqrt{2} = \{ q \in \mathbb{Q} \mid q < 0 \text{ or } q^2 < 2 \}. \]
Addition:
\[ A + B = \{ a + b \mid a \in A,\, b \in B \}. \]
Multiplication:
(for positive cuts) \[ A \times B = \{ a \times b \mid a \in A,\, b \in B,\; a,b > 0 \}. \]
Negatives and general cases are defined symmetrically by extending signs.
Order Relation:
Define: \[ A < B \iff A \subsetneq B. \]
This relation extends the usual order on \(\mathbb{Q}\) and makes \((\mathbb{R}, <)\) a totally ordered set.
Theorems:
\(a+b = b+a, \qquad a\times b = b\times a\).
\(a+(b+c) = (a+b)+c, \qquad a\times(b\times c) = (a\times b)\times c\).
\(a+0 = a, \qquad a\times1 = a\).
\(\forall a \in \mathbb{Q},\; \exists (-a) \text{ such that } a+(-a)=0\).
\(\forall a \in \mathbb{Q}\setminus\{0\},\; \exists a^{-1} \text{ such that } a\times a^{-1}=1\).
\(a\times(b+c) = a\times b + a\times c\).
The set of rational numbers \(\mathbb{Q}\) is dense in \(\mathbb{R}\):
\[ \forall a,b \in \mathbb{R},\; a < b \Rightarrow \exists r \in \mathbb{Q} \text{ such that } a < r < b. \]
1.2 Sequences and Convergent Sequences
Let \(\{a_n\}\) be a sequence and \(a\) be a real number. If for any given \(\varepsilon > 0\), there exists an \(N \in \mathbb{N}^*\) such that whenever \(n > N\), we have: \[ |a_n - a| < \varepsilon \]
then the sequence \(\{a_n\}\) is said to have the limit \(a\) as \(n\) tends to infinity, denoted as: \[ \lim_{n \to \infty} a_n = a \]
It can also be briefly denoted as \(a_n \to a\) (\(n \to \infty\)). We also say that the sequence \(\{a_n\}\) converges to \(a\). A sequence that has a limit is called a convergent sequence; a sequence that does not converge is called a divergent sequence.
Examples:
- \(\alpha > 0\), \(\lim\limits_{n \to \infty} \frac{1}{n^\alpha} = 0.\)
- \(\lim\limits_{n \to \infty} n^{1/n} = 1.\)
Theorem 1.1 If the sequence \(\{a_n\}\) converges, then it has only one limit. That is to say, the limit of a convergent sequence is unique.
Let \(\{a_n\}\) be a sequence. If there exists a real number \(A\), such that \(a_n \leqslant A\) holds for all \(n \in \mathbb{N}^*\), then \(\{a_n\}\) is said to be bounded above, and \(A\) is an upper bound of this sequence.
Similarly, we can define a sequence that is bounded below.
If the sequence \(\{a_n\}\) has both a lower bound and an upper bound, then it is called a bounded sequence.
Theorem 1.2 Convergent sequences are bounded.
Let \(\{a_n\}\) be a sequence, and let \(k_i \in \mathbb{N}^*\) (\(i=1, 2, 3, \dots\)) satisfy \(k_1 < k_2 < k_3 < \dots\). Then the sequence \(\{a_{k_n}\}\) is called a subsequence of \(\{a_n\}\).
Theorem 1.3 Let the limit of a convergent sequence \(\{a_n\}\) be \(a\). Then any subsequence of \(\{a_n\}\) also converges to \(a\).
Corollary 1.1 A necessary and sufficient condition for the sequence \(\{a_n\}\) to converge is that both its even subsequence \(\{a_{2n}\}\) and its odd subsequence \(\{a_{2n-1}\}\) converge, and have the same limit.
Theorem 1.4 Let \(\{a_n\}\) and \(\{b_n\}\) both be convergent sequences. Then \(\{a_n + b_n\}\) and \(\{a_n b_n\}\) are also convergent sequences. If \(\lim\limits_{n \to \infty} b_n \neq 0\), then \(\{a_n / b_n\}\) is also convergent, and:
\(\text{(1)}\quad \lim\limits_{n \to \infty} (a_n \pm b_n) = \lim\limits_{n \to \infty} a_n \pm \lim\limits_{n \to \infty} b_n.\)
\(\text{(2)}\quad \lim\limits_{n \to \infty} a_n b_n = \lim\limits_{n \to \infty} a_n \cdot \lim\limits_{n \to \infty} b_n.\)
\(\text{(3)}\quad \lim\limits_{n \to \infty} \frac{a_n}{b_n} = \frac{\lim\limits_{n \to \infty} a_n}{\lim\limits_{n \to \infty} b_n}, \quad \text{where } \lim\limits_{n \to \infty} b_n \neq 0.\)
If the limit of a convergent sequence \(\{a_n\}\) is equal to 0, then this sequence is called an infinitesimal sequence, or simply referred to as an infinitesimal.
Theorem 1.5 Let \[ a_n \leqslant b_n \leqslant c_n \quad (n \in \mathbb{N}^*) \]
If \(\lim\limits_{n \to \infty} a_n = \lim\limits_{n \to \infty} c_n = a\), then \[ \lim_{n \to \infty} b_n = a \]
Examples:
- \(a > 1, k \in \mathbb{N}^*\), \(\lim\limits_{n \to \infty}\frac{n^k}{a^n} = 0.\)
- \(\lim\limits_{n \to \infty} \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \cdots + \frac{1}{\sqrt{n^2+n}} = 1\).
If the sequence \(\{a_n\}\) satisfies the condition: for any positive number \(A\), there exists an \(N \in \mathbb{N}^*\) such that whenever \(n > N\), we have \(a_n > A\), then the sequence \(\{a_n\}\) is said to tend to \(+\infty\) (positive infinity), denoted as \[ \lim_{n \to \infty} a_n = +\infty. \]
If for any positive number \(A\), there exists an \(N \in \mathbb{N}^*\) such that whenever \(n > N\), we have \(a_n < -A\), then the sequence \(\{a_n\}\) is said to tend to \(-\infty\) (negative infinity), denoted as \[ \lim_{n \to \infty} a_n = -\infty. \]
If \(\lim\limits_{n \to \infty} |a_n| = +\infty\), then the sequence \(\{a_n\}\) is said to tend to \(\infty\), denoted as \(\lim\limits_{n \to \infty} a_n = \infty\).
Regardless of which of the three cases \[ \lim_{n \to \infty} a_n = +\infty, \quad \lim_{n \to \infty} a_n = -\infty, \quad \lim_{n \to \infty} a_n = \infty \] holds, the sequence \(\{a_n\}\) is called infinity (or an infinite sequence).
1.3 Monotonic Sequences
If the sequence \(\{a_n\}\) satisfies \[ a_n \leqslant a_{n+1} \quad (n = 1, 2, \dots), \] then this sequence is called an increasing sequence; if \(\{a_n\}\) satisfies \[ a_n \geqslant a_{n+1} \quad (n = 1, 2, \dots), \] then this sequence is called a decreasing sequence. If the above two inequalities are strict, i.e., \(a_n < a_{n+1}\) (or \(a_n > a_{n+1}\)) (\(n=1, 2, \dots\)), then this sequence is called strictly increasing (or strictly decreasing).
Increasing or decreasing sequences are collectively called monotonic sequences.
Theorem 1.6 A monotonic and bounded sequence must have a limit.
Theorem 1.7 For \(n \in \mathbb{N}^*\), let the sequence \(\{a_n\}\) be defined by: \[ a_n = 1 + \frac{1}{2^\alpha} + \cdots + \frac{1}{n^\alpha} \]
where \(\alpha > 1\). Then the sequence \(\{a_n\}\) is convergent.
Theorem 1.8 Let \(I_n = [a_n, b_n]\) (\(n \in \mathbb{N}^*\)), and let \[ I_1 \supset I_2 \supset I_3 \supset \cdots \supset I_n \supset I_{n+1} \supset \cdots. \]
If the length of this sequence of intervals \(|I_n| = b_n - a_n \to 0\) (\(n \to \infty\)), then the intersection \[ \bigcap_{n=1}^{\infty} I_n \] contains a unique point.
1.4 The Base of Natural Logarithm e
Let two sequences be defined as \[ a_n = \left(1 + \frac{1}{n}\right)^n, \quad b_n = \left(1 + \frac{1}{n}\right)^{n+1}. \]
Then:
- \(a_n\) is monotonically increasing.
- \(b_n\) is monotonically decreasing.
- For all \(n\), \(a_n < b_n\).
- Therefore, both sequences converge and have the same limit.
We define this common limit as
\[ e = \lim_{n \to +\infty} \left(1 + \frac{1}{n}\right)^n. \]
Theorem 1.9 Let the sequence \(\{s_n\}\) be defined as: \[ s_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} \quad (n \in \mathbb{N}^*). \]
Then for any positive integer \(n\), the error in approximating the base of the natural logarithm \(e\) by \(s_n\) satisfies the following inequality: \[ 0 < e - s_n \leqslant \frac{1}{n! n} \quad (n \in \mathbb{N}^*). \]
1.5 Fundamental Sequences and Cauchy Convergence Criterion
Let \(\{a_n\}\) be a real sequence. For any given \(\varepsilon > 0\), if there exists an \(N \in \mathbb{N}^*\) such that whenever \(m, n \in \mathbb{N}^*\) and \(m, n > N\), we have \[ | a_m - a_n | < \varepsilon, \] then the sequence \(\{a_n\}\) is called a fundamental sequence or a Cauchy sequence.
Lemma 1.1 Every sequence contains a monotonic subsequence.
Theorem 1.10 From any bounded sequence, a convergent subsequence can always be selected.
Theorem 1.11 A necessary and sufficient condition for a sequence to converge is that it is a fundamental sequence.
1.6 Supremum and Infimum
Let \(E\) be a non-empty set bounded above. A real number \(\beta\) satisfies the following two conditions:
\(\text{(1)}\quad\)For any \(x \in E\), we have \(x \leqslant \beta\);
\(\text{(2)}\quad\)For any given \(\varepsilon > 0\), there must exist an \(x_{\varepsilon} \in E\) such that \(x_{\varepsilon} > \beta - \varepsilon\).
In this case, \(\beta\) is called the supremum of the set \(E\), denoted as \(\beta = \sup E\).
Let \(E\) be a non-empty set bounded below. A real number \(\alpha\) satisfies the following two conditions:
\(\text{(1)}\quad\)For any \(x \in E\), we have \(x \geqslant \alpha\);
\(\text{(2)}\quad\)For any given \(\varepsilon > 0\), there must exist a \(y_{\varepsilon} \in E\) such that \(y_{\varepsilon} < \alpha + \varepsilon\).
In this case, \(\alpha\) is called the infimum of the set \(E\), denoted as \(\alpha = \inf E\).
Theorem 1.12 A non-empty set bounded above must have a supremum; a non-empty set bounded below must have an infimum.
Theorem 1.13 Let \([a, b]\) be a finite closed interval, and let \(\{I_\lambda\}\) be an open cover of it. Then a finite number of members (open intervals) can necessarily be selected from this family of open intervals, such that the family formed by these finite open intervals is still an open cover of \([a, b]\).
1.7 Limit Superior and Limit Inferior
Let \(\{a_n\}\) be a sequence, and let \(E\) be the set composed of all limit points of \(\{a_n\}\). Let \[ a^* = \sup E, \quad a_* = \inf E, \] then \(a^*\) and \(a_*\) are called the limit superior and limit inferior of the sequence \(\{a_n\}\), respectively, denoted as \[ a^* = \limsup_{n \to \infty} a_n, \quad a_* = \liminf_{n \to \infty} a_n. \]
Theorem 1.14 Let \(\{a_n\}\) be a sequence. Then:
\(\text{(1)}\quad a^* \in E\);
\(\text{(2)}\quad\)If \(x > a^*\), then there exists an \(N \in \mathbb{N}^*\) such that whenever \(n \geqslant N\), we have \(a_n < x\);
\(\text{(3)}\quad a^*\) is the unique number satisfying the first two properties.
Theorem 1.15 For the sequence \(\{a_n\}\), define \(\alpha_n = \inf\limits_{k \geqslant n} a_k\) and \(\beta_n = \sup\limits_{k \geqslant n} a_k\). Then:
\(\text{(1)}\quad \{\alpha_n\}\) is an increasing sequence, and \(\{\beta_n\}\) is a decreasing sequence;
\(\text{(2)}\quad \lim\limits_{n \to \infty} \alpha_n = a_*, \quad \lim\limits_{n \to \infty} \beta_n = a^*\).
1.7.1 Stolz Theorem
Theorem 1.16 (\(\frac{\infty}{\infty}\) Type) Let \(\{b_n\}\) be a strictly increasing sequence tending to \(+\infty\). If \[ \lim_{n \to \infty} \frac{a_n - a_{n-1}}{b_n - b_{n-1}} = A, \]
then \[ \lim_{n \to \infty} \frac{a_n}{b_n} = A, \]
where \(A\) can be \(+\infty\) or \(-\infty\).
Theorem 1.17 (\(\frac{0}{0}\) Type) Let \(\{a_n\}\) and \(\{b_n\}\) be two sequences converging to 0. If \(\{b_n\}\) is strictly decreasing and \[ \lim_{n \to \infty} \frac{a_n - a_{n-1}}{b_n - b_{n-1}} = A, \]
then \[ \lim_{n \to \infty} \frac{a_n}{b_n} = A, \]
where \(A\) can be a finite number, \(+\infty\), or \(-\infty\).