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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
- Definition: Real Sequence
- Definition: Convergent Real Sequence
- Definition: Divergent Sequences
- Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
- Definition: Monotonic Sequences
- Section: Theorems Regarding Limits Of Sequences
- Definition: Limits of Real Functions
- Section: Theorems Regarding Limits of Functions
- Section: Examples of Limit Calculations
- Definition: Real Subsequence
- Theorem: Every Bounded Real Sequence has a Convergent Subsequence
- Definition: Accumulation Point (Real Numbers)
- Definition: Asymptotical Approximation
- Lemma: Decreasing Sequence of Suprema of Extended Real Numbers
- Definition: Limit Superior
- Lemma: Increasing Sequence of Infima of Extended Real Numbers
- Definition: Limit Inferior
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