C. Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.Hint: As usual, there are two directions to prove in an ⇔. The picture on p73 of the notes may be somewhat helpful. Proportions Consider a sequence (sn) of points in the set IR²We have Lim Sn = Sin(IR², d₂)if and only if lim S=5(IR², d.),Self-chaks Adjust the proof that limⒸlin asaan + b₂i=a+bib₂=b₁Geometrically, this holle because we can put a small"doc" w/ {* radingwrt d, inside an"docw/ EradiesWe have lim Sn=Slimit of parts MMinExcumplai Given a sequence of partswrt dzDistancesare real numbers in every matriz space.We have the following relation between limits in M and limits i PR.Propunition: Let (sa) be asequenceof points in metric space (M4, d).if and only if lim d (sn₁ st = 0.Clint ink1-3Example!!;Canthen73Profs Observe that the two definitions are the same up to eary algebra,[n>N] = [ d[sa, s)<{]VESO, IN St.• VE>O, AN st. [n>1] → [Id(ss)<ε],and vice-versa.Sn in M, if youshowthat for each , d (sas) = Y₂"you have shrom Lusing the Prop.) that lim sa=s. ✓If z is a complex number with 12/<1,then lim 121"=0, and 12²-01=121",Thus, lim 2"=0,

Euclidean metric dEuc((a,b),(p,q))=√((a-p)2+(b-q)2) Maximum metric dmax((a,b),(p,q))=max{|a-p| ,…

Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As usual, there are two directions to prove in an ⇔. The picture on p73 of the notes may be somewhat helpful.

Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric. Hint: As usual, there are two directions to prove in an ⇔. The picture on p73 of the notes may be somewhat helpful.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

C. Show that a set A in ℝ² is open in the Euclidean metric ⇔ it is open in the max metric.
Hint: As usual, there are two directions to prove in an ⇔. The picture on p73 of the notes may be somewhat helpful.

Proportions Consider a sequence (sn) of points in the set IR²
We have Lim Sn = S
in
(IR², d₂)
if and only if lim S=5
(IR², d.),
Self-chaks Adjust the proof that lim
Ⓒlin asa
an + b₂i=a+bi
b₂=b₁
Geometrically, this holle because we can put a small
"doc" w/ {* rading
wrt d, inside an
"doc
w/ E
radies
We have lim Sn=S
limit of parts MM
in
Excumplai Given a sequence of parts
wrt dz
Distances
are real numbers in every matriz space.
We have the following relation between limits in M and limits i PR.
Propunition: Let (sa) be a
sequence
of points in metric space (M4, d).
if and only if lim d (sn₁ st = 0.
Clint ink
1-3
Example!!
;
Can
then
73
Profs Observe that the two definitions are the same up to eary algebra,
[n>N] = [ d[sa, s)<{]
VESO, IN St.
• VE>O, AN st. [n>1] → [Id(ss)<ε],
and vice-versa.
Sn in M, if you
show
that for each , d (sas) = Y₂"
you have shrom Lusing the Prop.) that lim sa=s. ✓
If z is a complex number with 12/<1,
then lim 121"=0, and 12²-01=121",
Thus, lim 2"=0,

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