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