Question. (math analysis) (X ,d ) is a metric space and (p_n )_(n=1) ^∞ is a sequence in X. (p_n )_(n=1) ^( ∞) is cauchy if and only if lim_(N→∞) diam (E_N )=0. where , E_N = { p_N , p


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Question Number 205559 by mnjuly1970 last updated on 24/Mar/24
$Question. (math analysis) (X ,d ) is a metric space and (p_n )_(n=1) ^∞ is a sequence in X. (p_n )_(n=1) ^( ∞) is cauchy if and only if lim_(N→∞) diam (E_N )=0. where , E_N = { p_N , p_(N+1) , ...} diam E:=sup{d(x,y)∣x,y ∈E }$
$Q u e s t i o n . (m a t h a n a l y s i s)$ $(X, d) i s a m e t r i c s p a c e a n d$ ${(p_{n})}_{n = 1}^{\infty} i s a s e q u e n c e i n X .$ ${(p_{n})}_{n = 1}^{\infty} i s c a u c h y i f a n d o n l y i f$ $\lim_{N \to \infty} d i a m (E_{N}) = 0 .$ $w h e r e, E_{N} = {p_{N}, p_{N + 1}, . . .}$ $d i a m E := s u p {d (x, y) ∣ x, y \in E}$
	Answered by Berbere last updated on 24/Mar/24
	$dim(E)=sup d(x_i ,x_j );(x_i ,x_j )∈E^2 Let p_(n ) a cauchy sequence⇒∀ε>0 ∃N∈N ∣∀k,m≥0 d(P_(N+k) ,P_(N+m) )≤ε......1 E_N {P_N ,P_(N+1) ,.......};let S_N =sup (d(x,y)∣(x,y)∈E_N ^2 ) S_N ∈R since P_N is cauchy Sequence ∃a>0 ⇒d(P_k ,P_l )<a ∀(k,l)∈N^2 if N_1 >N_2 ⇒E_N_2 ⊂E_N_1 ⇒S_N_2 <S_N_1 S_(N ) decrease biunded Sequence ⇒Diam(E_N ) Cv 1...⇒∀ε≥0 ∃N ∈N ∣ diam(E_N )<ε lim_(N→∞) diam(E_N )=a∈R_+ a<diam(E_N ) (E_N decrease)⇒∀ε>0 0≤ a<ε ∀n>0 0≤a≤(1/n)⇒a∈∩^(n∈N^∗ ) [0,(1/n)]={0} ⇒lim_(n→∞) diam(E_N )=0$
	$d i m (E) = s u p d (x_{i}, x_{j}); (x_{i}, x_{j}) \in E^{2}$ $L e t p_{n} a c a u c h y s e q u e n c e \Rightarrow \forall ϵ > 0 \exists N \in N ∣ \forall k, m ⩾ 0 d (P_{N + k}, P_{N + m}) ⩽ ϵ . . . . . . 1$ $E_{N} {P_{N}, P_{N + 1}, . . . . . . .}; l e t S_{N} = s u p (d (x, y) ∣ (x, y) \in E_{N}^{2})$ $S_{N} \in R s i n c e P_{N} i s c a u c h y S e q u e n c e \exists a > 0 \Rightarrow d (P_{k}, P_{l}) < a$ $\forall (k, l) \in N^{2}$ $i f N_{1} > N_{2} \Rightarrow E_{N_{2}} \subset E_{N_{1}} \Rightarrow S_{N_{2}} < S_{N_{1}} S_{N} d e c r e a s e b i u n d e d S e q u e n c e$ $\Rightarrow D i a m (E_{N}) C v$ $1 . . . \Rightarrow \forall ϵ ⩾ 0 \exists N \in N ∣ d i a m (E_{N}) < ϵ$ $\lim_{N \to \infty} d i a m (E_{N}) = a \in R_{+}$ $a < d i a m (E_{N}) (E_{N} d e c r e a s e) \Rightarrow \forall ϵ > 0 0 ⩽ a < ϵ$ $\forall n > 0 0 ⩽ a ⩽ \frac{1}{n} \Rightarrow a \in \overset{n \in N^{*}}{\cap} [0, \frac{1}{n}] = {0}$ $\Rightarrow \lim_{n \to \infty} d i a m (E_{N}) = 0$
	Commented by mnjuly1970 last updated on 24/Mar/24

	$t h a n k s a l o t s i r . . .$
	Commented by Berbere last updated on 24/Mar/24

	$W i t h e P l e a s u r$
	Answered by aleks041103 last updated on 04/Apr/24
	$Let a_N =diam({p_k }_(k=N) ^∞ ). We are asked to prove {p_k }_(k=1) ^∞ is Cauchy ⇔ lim_(N→∞) a_N = 0 1. Cauchy ⇒ a_N →0 Cauchy ⇒ ∀ε>0,∃M:d(p_m ,p_n )<ε/2, ∀m,n>M ⇒a_M =diam({p_k }_(k=M+1) ^∞ )=sup{d(p_m ,p_n )∣m,n>M}≤ε/2<ε it is easy to see that a_n >0 is decreasing ⇒∀m≥M, 0<a_m ≤a_M <ε, i.e. ∣a_m ∣<ε ⇒∀ε>0,∃M:∣a_m −0∣<ε, ∀m>M ⇒lim_(N→∞) a_N =0 2.a_n → 0 ⇒ Cauchy a_N → 0 (a_N >0)⇒ ∀ε>0, ∃M: a_N <ε, ∀N≥M ⇒a_M =sup_(m,n≥M) d(p_m ,p_n ) < ε ⇒d(p_m ,p_n )<ε, ∀m,n>M ⇒∀ε>0,∃M:d(p_m ,p_n )<ε, ∀m,n>M ⇒Cauchy$
	$L e t a_{N} = d i a m ({p_{k}}_{k = N}^{\infty}) .$ $W e a r e a s k e d t o p r o v e$ ${p_{k}}_{k = 1}^{\infty} i s C a u c h y \Leftrightarrow \underset{N \to \infty}{l i m} a_{N} = 0$ $1 . C a u c h y \Rightarrow a_{N} \to 0$ $C a u c h y \Rightarrow \forall ϵ > 0, \exists M : d (p_{m}, p_{n}) < ϵ / 2, \forall m, n > M$ $\Rightarrow a_{M} = d i a m ({p_{k}}_{k = M + 1}^{\infty}) = s u p {d (p_{m}, p_{n}) ∣ m, n > M} ⩽ ϵ / 2 < ϵ$ $i t i s e a s y t o s e e t h a t a_{n} > 0 i s d e c r e a s i n g$ $\Rightarrow \forall m ⩾ M, 0 < a_{m} ⩽ a_{M} < ϵ, i . e . ∣ a_{m} ∣< ϵ$ $\Rightarrow \forall ϵ > 0, \exists M :∣ a_{m} - 0 ∣< ϵ, \forall m > M$ $\Rightarrow \underset{N \to \infty}{l i m} a_{N} = 0$ $2 . a_{n} \to 0 \Rightarrow C a u c h y$ $a_{N} \to 0 (a_{N} > 0) \Rightarrow \forall ϵ > 0, \exists M : a_{N} < ϵ, \forall N ⩾ M$ $\Rightarrow a_{M} = \underset{m, n ⩾ M}{s u p} d (p_{m}, p_{n}) < ϵ$ $\Rightarrow d (p_{m}, p_{n}) < ϵ, \forall m, n > M$ $\Rightarrow \forall ϵ > 0, \exists M : d (p_{m}, p_{n}) < ϵ, \forall m, n > M$ $\Rightarrow C a u c h y$
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