Teacher: Liubiyu

Chapter 1-2

Contents §1.2 Elementary functions and graph §2.1 Limits of Sequence of number §2.2 Limits of functions §1.1 Sets and the real number §2.3 The operation of limits §2.4 The principle for existence of limits §2.5 Two important limits §2.6 Continuity of functions §2.7 Infinitesimal and infinity quantity, the order for infinitesimals

New words 闭区间套定理 The Nested Interval Theorem 闭区间套定理 The Bolzano-Weierstrass Theorem 夹逼原理 The Squeeze Principle 夹逼原理 The Monotone and Bounded Principle单调有界原理

§2.4 The principle for existence of limits It has been pointed out that the definition of limit does not give any methods for finding the limit of a given function, and does not even offer a criterion to judge whether the limit of given function exist. Hence, after we have learned the definition of limit, it is important to find conditions for existence of the limit of a function. 1 、 The two important principles Theorem 1 Theorem 1 ( The Squeeze Principle ) If the following conditions are satisfied

Proof Proof

According to the precise definition of limits, we have Similarly, we have Corollary 1

Notations (1) According to this corollary 1, we obtain another method to prove the convergence of a given sequence and find its limit at the same time. Making each terms of the given sequence large and smaller we get two new sequences respectively. If the limits of these two new sequences are the same value A, then the limiting of the given sequence is also A.A.

Example 1 Solution According to theorem 1, we have

Example 2 Proof

Definition 1 Definition 1 ( Monotonicity of a sequence )

Theorem 2 Theorem 2 (The Monotone and Bounded Principle) Proof Proof

From the inequalities (2) and (3), we obtain the required inequality (1). Thus,

Notations (2) It is easy to see that changing any finite number of terms of a sequence does not affect its convergence and limit. Therefore, if a sequence is bounded but monotone only starting from some term, then it is still convergent.

Example 3 Solution

2 、 The Nested Interval Theorem Theorem 3 Theorem 3 ( The nested interval theorem )

See the following figure Example 4 Proof

which completes the proof.

3 、 The Cauchy Convergence Principle Theorem 4 Theorem 4 (The Cauchy Convergence Principle) Definition 2

Theorem 5 Theorem 5 (The Bolzano-Weierstrass Theorem) Every bounded sequence has a convergent subsequence