Write β₯ β β
β β₯ \lVert\,\cdot\,\rVert β₯ β
β₯ Euclidean norm , z + z β² z+z' z + z β² Ξ» z \lambda z Ξ» z sum and the scalar multiple , and z β z β² z-z' z β z β² difference of points of R n \mathbb{R}^n R n β£ β
β£ |\cdot| β£ β
β£ absolute value on R \mathbb{R} R
Claim 1. Let u , v β B Λ d E ( x , r ) u,v\in\bar{B}_{d_E}(x,r) u , v β B Λ d E β β ( x , r ) t β R t\in\mathbb{R} t β R 0 β€ t 0\le t 0 β€ t t β€ 1 t\le1 t β€ 1 Elementary Properties of the Euclidean Norm on R n \mathbb{R}^n R n , d E ( x , u ) = β₯ x β u β₯ d_E(x,u)=\lVert x-u\rVert d E β ( x , u ) = β₯ x β u β₯ d E ( x , v ) = β₯ x β v β₯ d_E(x,v)=\lVert x-v\rVert d E β ( x , v ) = β₯ x β v β₯ β₯ x β u β₯ β€ r \lVert x-u\rVert\le r β₯ x β u β₯ β€ r β₯ x β v β₯ β€ r \lVert x-v\rVert\le r β₯ x β v β₯ β€ r
In the real vector space R n \mathbb{R}^n R n Euclidean Space R n \mathbb{R}^n R n we have t β x + ( 1 β t ) β x = x t\,x+(1-t)\,x=x t x + ( 1 β t ) x = x
x β ( t β u + ( 1 β t ) β v ) = t β ( x β u ) + ( 1 β t ) β ( x β v ) . x-\bigl(t\,u+(1-t)\,v\bigr)=t\,(x-u)+(1-t)\,(x-v). x β ( t u + ( 1 β t ) v ) = t ( x β u ) + ( 1 β t ) ( x β v ) . By claim 6 of Elementary Properties of the Euclidean Norm on R n \mathbb{R}^n R n and then claim 5 of that lemma,
β₯ x β ( t β u + ( 1 β t ) β v ) β₯ β€ β£ t β£ β β₯ x β u β₯ + β£ 1 β t β£ β β₯ x β v β₯ . \bigl\lVert x-\bigl(t\,u+(1-t)\,v\bigr)\bigr\rVert\le|t|\,\lVert x-u\rVert+|1-t|\,\lVert x-v\rVert . β x β ( t u + ( 1 β t ) v ) β β€ β£ t β£ β₯ x β u β₯ + β£1 β t β£ β₯ x β v β₯ . We have 0 β€ t 0\le t 0 β€ t 0 β€ 1 β t 0\le 1-t 0 β€ 1 β t Elementary Arithmetic in an Ordered Field applied to t β€ 1 t\le1 t β€ 1 β£ t β£ = t |t|=t β£ t β£ = t β£ 1 β t β£ = 1 β t |1-t|=1-t β£1 β t β£ = 1 β t Absolute Value in an Ordered Field . Applying claim 5 of Elementary Arithmetic in an Ordered Field to β₯ x β u β₯ β€ r \lVert x-u\rVert\le r β₯ x β u β₯ β€ r t t t β₯ x β v β₯ β€ r \lVert x-v\rVert\le r β₯ x β v β₯ β€ r 1 β t 1-t 1 β t Elementary Arithmetic in an Ordered Field ,
t β β₯ x β u β₯ + ( 1 β t ) β β₯ x β v β₯ β€ t β r + ( 1 β t ) β r = r , t\,\lVert x-u\rVert+(1-t)\,\lVert x-v\rVert\le t\,r+(1-t)\,r=r, t β₯ x β u β₯ + ( 1 β t ) β₯ x β v β₯ β€ t r + ( 1 β t ) r = r , the last equality by distributivity, since t + ( 1 β t ) = 1 t+(1-t)=1 t + ( 1 β t ) = 1 β€ \le β€ d E ( x , β t β u + ( 1 β t ) β v ) β€ r d_E\bigl(x,\,t\,u+(1-t)\,v\bigr)\le r d E β ( x , t u + ( 1 β t ) v ) β€ r t β u + ( 1 β t ) β v β B Λ d E ( x , r ) t\,u+(1-t)\,v\in\bar{B}_{d_E}(x,r) t u + ( 1 β t ) v β B Λ d E β β ( x , r ) u u u v v v t t t convex .
Claim 2. By claim 3 of Elementary Properties of the Closed Ball in a Metric Space , B Λ d E ( x , r ) \bar{B}_{d_E}(x,r) B Λ d E β β ( x , r ) closed subset of ( R n , T d E ) (\mathbb{R}^n,\mathcal{T}_{d_E}) ( R n , T d E β β ) The Closure is the Smallest Closed Superset its closure in that topological space is B Λ d E ( x , r ) \bar{B}_{d_E}(x,r) B Λ d E β β ( x , r ) Elementary Properties of the Closed Ball in a Metric Space it is bounded in ( R n , d E ) (\mathbb{R}^n,d_E) ( R n , d E β ) . Hence The Closure of a Bounded Subset of R n \mathbb{R}^n R n , applied to this bounded set, shows that B Λ d E ( x , r ) \bar{B}_{d_E}(x,r) B Λ d E β β ( x , r ) ( R n , T d E ) (\mathbb{R}^n,\mathcal{T}_{d_E}) ( R n , T d E β β )