# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 103: Dedekind Cuts

The Dedekind cut is a method of constructing $\mathbb{R}$ from $\mathbb{Q}$. A cut is first defined to be a partition of $\mathbb{Q}$ into 2 non-empty subsets $A, B$ such that

1. if $p \in A$ and $q < p$, then $q \in A$
2. if $p \in A$, then there is a $q \in A$ such that $q > p$ (meaning $A$ has no maximum element)

A cut can be equivalently determined solely by $A$ alone, rather than the pair $(A, B)$.

Cuts can then be used to construct $\mathbb{R}$ by defining any $r \in \mathbb{R}$ to be the cut where $A$ is the set of all members $q$ of $\mathbb{Q}$ such that $q < r$. That is, $r$ is simply defined as the subset of rationals smaller than itself, and $\mathbb{R}$ is the set of all such subsets.

## 97: The Archimedean Property of the Reals

$\mathbb{R}$ has the Archimedean property, which states that for any positive $x, y \in \mathbb{R}$ there exists an $n \in \mathbb{N}$ such that

$nx > y$.

Intuitively, this means that $\mathbb{R}$ contains neither infinitely large nor infinitesimally small numbers. (The property would not hold if $y$ was infinite, or if $x$ was infinitesimal.)

#### Proof:

Finding an $n$ such that $nx > y$ is equivalent to finding one such that $n > \frac{y}{x}$. If there is no such $n$, this means that $n \leq \frac{y}{x}$ for all $n$ and thus the number $\frac{y}{x}$ is an upper bound on $\mathbb{N}$. However, $\mathbb{N}$ has no upper bound, so this is a contradiction, meaning such an $n$ must exist.

## 58: Convergence of Geometric Series with Complex Ratios

Where $\lambda \in \mathbb{C}$,  $\lim _{n \to \infty} \lambda ^n$ exists if and only if $\lambda = 1$ or $|\lambda | < 1$.