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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 67E
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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,
Transcribed Image Text: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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