Again, for v € R²\ {0}, define the open half line D, through {0} defined D₂ = {tv|t>0} CR². Consider a sequence of vectors Un E R², n ≥ 1 such that Un → Vo 0. For example Un = (1, 1/n) → V∞ = (1,0). by V D D 1 D D a) Show that the set R²\(Un>1D) is a neighborhood of every point in R²\({0} U Dv U (Un>1Dvn)); but is not a neighborhood of any point in {0}UD U (Un>1Dvn). b) Show that neither Un>1Den nor {0} U UniDen is a closed subset of R². c) Show that {0} U Dv U (Un>Don) is a closed subset of R².
Again, for v € R²\ {0}, define the open half line D, through {0} defined D₂ = {tv|t>0} CR². Consider a sequence of vectors Un E R², n ≥ 1 such that Un → Vo 0. For example Un = (1, 1/n) → V∞ = (1,0). by V D D 1 D D a) Show that the set R²\(Un>1D) is a neighborhood of every point in R²\({0} U Dv U (Un>1Dvn)); but is not a neighborhood of any point in {0}UD U (Un>1Dvn). b) Show that neither Un>1Den nor {0} U UniDen is a closed subset of R². c) Show that {0} U Dv U (Un>Don) is a closed subset of R².
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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Exercise 3
Part c only!
Transcribed Image Text:Again, for v E R² \ {0}, define the open half line D, through {0} defined
D₂ = {tv|t>0} CR².
Consider a sequence of vectors Un E R², n ≥ 1 such that Un v0. For example
Un = (1, 1/n) → v∞ = (1,0).
by
3
1₂
N₂
D₂₁
Dv₂
1
Dy
Du
a) Show that the set R²\(Unz1 Dun) is a neighborhood of every point in R²\({0} U Dv U (Un≥1Dvn)),
but is not a neighborhood of any point in {0}UD U (Un≥1 Dvn).
b) Show that neither Unz1D nor {0} U Un>1Dvn
{0} U Dv U (Un>1Dun) is a closed subset of R².
is a closed subset of R2. c) Show that
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