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Real-valued Sequences and Limits of Sequences and Functions

In this chapter, we study the behavior of the sequences in the field of real numbers $(\mathbb R, +, \cdot)$ but most of the theorems and definitions will be also applicable for the field of rational numbers $(\mathbb Q, +, \cdot),$ and also the field of complex numbers $(\mathbb C, +, \cdot).$ Whenever this is the case, we will indicate it by a marker like this:

applicability: $\mathbb {Q, R, C}$

Sometimes, also the sets of natural numbers $\mathbb N$ or of integers $\mathbb Z$ will be listed, if a definition is applicable also for them, exploring the set inclusion $\mathbb N\subset \mathbb Z\subset\mathbb Q\subset\mathbb R.$

| | | | created: 2014-02-20 22:23:24 | modified: 2020-07-10 18:27:56 | by: bookofproofs

1.Definition: Real Sequence

2.Definition: Convergent Real Sequence

3.Definition: Divergent Sequences

4.Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence

5.Definition: Monotonic Sequences

6.Theorems Regarding Limits Of Sequences

7.Definition: Limits of Real Functions

8.Theorems Regarding Limits of Functions

9.Examples of Limit Calculations

10.Definition: Real Subsequence

11.Theorem: Every Bounded Real Sequence has a Convergent Subsequence (Bolzano, Weierstrass)

12.Definition: Accumulation Point (Real Numbers)

13.Definition: Asymptotical Approximation

14.Lemma: Decreasing Sequence of Suprema of Extended Real Numbers

15.Definition: Limit Superior

16.Lemma: Increasing Sequence of Infima of Extended Real Numbers

17.Definition: Limit Inferior

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Bibliography (further reading)