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Chapter: 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 $\mathbb N\subset \mathbb Z\subset\mathbb Q\subset\mathbb R.$

Table of Contents

  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. Section: Theorems Regarding Limits Of Sequences
  7. Definition: Limits of Real Functions
  8. Section: Theorems Regarding Limits of Functions
  9. Section: Examples of Limit Calculations
  10. Definition: Real Subsequence
  11. Theorem: Every Bounded Real Sequence has a Convergent Subsequence
  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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