## Existence of limits and completeness of the real numbers system

- Monotonicity and its implications.
- Nested intervals and their common point
- Boundedness as another property stable by finite alterations
- Converging subsequense of a bounded sequence
- But why we are so sure that there are no gaps on the real line? And what is a real line?

Construction of the number system: from natural numbers toward scary numbers

- Completion by algebraic operations: from to via . Everything you need to solve linear equations
- Problems with quadratic equations: irrationalities and negative discriminants. An idea of algebraic number.
- Problems with transition to limit: the ubiquitous and much, much more
- Infinite decimal fractions: completion by “adding limits of monotone sequences”.
- Operations with real numbers: ordered field.
**Completeness**“axiom”.

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