# Sergei Yakovenko's blog: on Math and Teaching

## 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 $\mathbb N$ to $\mathbb Q$ via $\mathbb Z$. 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 $\pi$ and much, much more
• Infinite decimal fractions: completion by “adding limits of monotone sequences”.
• Operations with real numbers: ordered field. Completeness “axiom”.