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. Cantionary tales Although sn= (-11"Indeed, sincethis isExamplesExampleDefinitions We saywhile anotherThis situationwithWe introduce terminology!thatishas convergent subsequencessubsequenosConsoConverse to -1,another prout thatThetheOnemetriz(-)"(-1)".nDefinition If a subsequence ofdive-sestorealandi) has an40ii)has,A(-11" is important and interesting,aspaceras anaccumulation print if r=lim Snk for somesubsequence (S) of (su).sayLemma A recl-valmed.not convergent, itZKSequence(M,d) hasA(sn) has(-1)(-1)" is dit-gent,and (-1) 2KHto 1 Las a constant sey)(su) of points in1anLor anreal-valued sequence (su)(or to-co),accumulation points at 1,-1AL4476accumulation point at co"40at co, -00,put ofi^is thesequearehas accumulation points at 1,0-1,4(1, 0,-1,0,1,...)Accumulation points at I wo give us an alternative wayto talk about unbunded sequences, as the following lemma tells us.(S₂)sequenceaccumulatim punt at a.).-Co(sa) has no bound from above, and(sa) has no bund from below. Profi First, noticekandSo it suffices to pone (i). We verify (=) and (5).As asubsequenceNowIt follows thatدی77that (ii) fullows from (i) by multiplying by -1;=co) lim-S₁ ===SoSuxVM kw >> KW/S₁ <-M.natural(The set of suchE): Recursively define the increasing subsequenceby n₁ = 1, and soNICHam(as 3e walSmokINSme diveger to co[K>N] =>{nx >M.arbitrary M is not in bond from above for forSme>M.)(strate))with neers > Moe and SMKk→numbusis well-defirdSukhr of INthat he is the leastisand unbranded as= cois nunempty sinceincreasing by construction"a)Somax(KH, Sn₂)Sn is unlunded,IN is well-ordered).Or, just takeN=Min limitdefinition!Sny > MeSn has accumulation pt at co.

Euclidean metric dEuc((a,b),(c,d))=√((a-c)2+(b-d)2) Maximum metric dMax((a,b),(c,d))=max{|a-c| ,…

Answered: Show that a set A in ℝ2 is open in the…

Show that a set A in ℝ2 is open in the Euclidean metric ⇔ it is open in the max metric.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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