In this thesis we take a look at the similarities and differences between Cauchy and quasi-Cauchy sequences. We start with the properties of quasi-Cauchy sequences in ${\mathbb R}^n$. After that we examine the continuity of mappings in terms of preserving Cauchy and quasi-Cauchy sequences. Later, we focus on quasi-Cauchy sequences in general metric spaces. We introduce the notion of nonincremental metric spaces. Of our special interest are ultrametric spaces, as it turns out that they are nonicremental. We take a look at the $p$-adic numbers as an interesting example of ultrametric spaces. Finally, we make a characterization of ultrametric spaces.
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