Open Ball Contains Open Ball Less Than Half Its Radius

Therorem

Let $M = \struct {A, d}$ be a metric space.

Let $a, b \in A$.

Let $\epsilon, \delta \in \R_{>0}$ such that $\delta \le \dfrac \epsilon 2$.

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball on $a$.

Let $\map {B_\delta} b$ be the open $\delta$-ball on $b$.

Let $a \in \map {B_\delta} b$.

Then:

$\map {B_\delta} b \subseteq \map {B_\epsilon} a$

We have:

$\ds \forall c \in \map {B_\delta} b: \, $

$\ds \map d {a, c}$

$\le$

$\ds \map d {a, b} + \map d {b, c}$

$\ds $

$<$

$\ds \delta + \map d {b, c}$

Definition of open $\delta$-ball and $a \in \map {B_\delta} b$

$\ds $

$<$

$\ds \delta + \delta$

Definition of open $\delta$-ball and $c \in \map {B_\delta} b$

$\ds $

$\le$

$\ds \dfrac \epsilon 2 + \dfrac \epsilon 2$

as $\delta \le \dfrac \epsilon 2$

$\ds $

$=$

$\ds \epsilon$

$\ds \leadsto \ \ $

$\ds \forall c \in \map {B_\delta} b: \, $

$\ds c$

$\in$

$\ds \map {B_\epsilon} a$

$\ds \leadsto \ \ $

$\ds \map {B_\delta} b$

$\subseteq$

$\ds \map {B_\epsilon} a$

Definition of Subset

$\blacksquare$