Sequences and limits

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Chapter 5 Sequences and limits

Sequences and their limits are fundamental notions in the study of real numbers. In this chapter, we will be interested in the the study of sequences and their limits (and in particular, when such limits exist).

We will develop rules to manipulate limits, and we will use this tool to construct roots of certain simple equations.

If we are given a sequence \((x_{n})\) that we suspect is convergent, but we don't have a candidate for the limit, we can nevertheless confirm, in certain cases, that a limit exists. The key tool is the Monotone Convergence Theorem, which we will prove here: it states that if \((x_{n})\) is increasing and bounded above, then it converges. This gives us a machine to construct a big collection of new real numbers.