Abhijeet Mulgund's Personal Webpage

Search

Cauchy Sequence

Last updated Nov 1, 2022

# Definition

Let $(M, d)$ be a Metric Space

. We say $({x}{n}){n=1}^{\infty} \subset M$ is a Cauchy Sequence if

\forall \epsilon> 0

ther exists

N \in \mathbb{N}

so that for all

n,m \geq N

,

d(x_{n}, x_{m}) < \epsilon

.

# Other Outlinks

Natural Numbers
Sequence

Sequence

Definition A $f: \mathbb{N} \to X$ for some $X$ . It is usually denoted $\{xn\}{n=1}^{\infty} \subset X$ or $(x{n}) \subset...

11/7/2022
Real Numbers

Real Numbers

Definition 1 The unique Complet totally ordere . It is usually denoted $\mathbb{R}$ and is used to represent the continuum. Remarks One...

11/7/2022

Backlinks

Cauchy Sequence

Cauchy Sequence

Definition Let $(M, d)$ be a . We say $({x}{n}){n=1}^{\infty} \subset M$ is a if $\forall \epsilon> 0$ ther exists $N \in \mathbb{N}$ so that for all $n,m \geq N$ , $d(x{n}, x{m}) < \epsilon$ . Other Outlinks ...

11/7/2022
Complete Metric Space

Complete Metric Space

Definition Let $(M, d)$ be a . We say $M$ is a if for all s $({x}{n}){n=1}^{\infty} \subset M$ , there exists $x \in M$ so that $x_{n} \to x$ . Other Outlinks ...

11/7/2022
Completeness of the Real Numbers

Completeness of the Real Numbers

...satisfies the $\mathbb{R}$ satisfies the . $\mathbb{R}$ satisfies the . $\mathbb{R}$ satisfies the Completing $\mathbb{R}$ Approach 1: Completion by s In this approach we define a notion of s on $\mathbb{Q}$ and represent $\mathbb{R}$ as s on......

11/7/2022
Convergent Sequences are Cauchy

Convergent Sequences are Cauchy

...\in \mathbb{N} $so that$ \forall n \geq N $,$ d(x{n}, x) < \frac{\epsilon}{2} $. Then$ \forall n,m \geq N $,$ $d(x{n},x{m}) \leq d(x{n}, x) + d(x, x{m}) < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon,$ $so$ ({x}{n}){n=1}^{\infty}$ is a . Other Outlinks ......

11/7/2022
Rationals are Dense in the Reals

Rationals are Dense in the Reals

...\in \mathbb{R}$ is defined as a an over s in $\mathbb{Q}$ , where elements of $\mathbb{Q}$ are identified with constant sequences. The space is endowed with the . Let $x \in \mathbb{R}$ be identified by the $({x}{n}){n=1}^{\infty} \subset \mathbb{Q}$ .......

11/7/2022

Interactive Graph