Today I Learned

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

Category Archives: analysis

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.