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Chapter: Realvalued 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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