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(9) Show that the given collection F is an open cover for S such that it does not contain a finite subcover and so s not compact. S = (0, 2); and F = {Un | neN} where Un = (1, 2-¹) (b) S (0, 0); and F = {Un | neN} where Un = (0, n) (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ [x₁ - x₂], [y₁ - yal}. Verify that this is a metric on R2 and for > 0 draw an arbitrary E-neighborhood for a point (x, y) = R². opert

(9) Show that the given collection F is an open cover for S such that it does not contain a finite subcover and so s not compact. S = (0, 2); and F = {Un | neN} where Un = (1, 2-¹) (b) S (0, 0); and F = {Un | neN} where Un = (0, n) (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ [x₁ - x₂], [y₁ - yal}. Verify that this is a metric on R2 and for > 0 draw an arbitrary E-neighborhood for a point (x, y) = R². opert

Advanced Engineering Mathematics
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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Ans. no. 9 only

(7) Let S = { 1 − (−1)” | n € N}. Determine inf S and sup S. Justify. n (8) Find the infimum, minimum, maximum and supremum of each of the following sets (write d.n.e. if it does not exist). S inf S min S max S sup S 1 1 - ² | neN} n {xER | x³-4x²+x+6 > 0 } = - = | m, n € N m n ∞ 1 2+ n n n=1 {XEQ x ≤ 2} (9) Show that the given collection F is an open cover for S such that it does not contain a finite subcover and so Sis not compact. - (a) S = (0, 2); and F = { Un | n E N} where Un. (-1,2-) (b) S = (0, ∞); and F = { Un | n E N} where Un = (0, n) 4 (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ x₁ - x₂l|y₁ - yal}. Verify that this is a metric on R² and for > 0, draw an arbitrary e-neighborhood for a point (x, y) E R². Success never rests. On your worst days, be good; On your best days, be great, and on every other day, be better. Carmen Mariano ор of PUR DO
Transcribed Image Text:(7) Let S = { 1 − (−1)” | n € N}. Determine inf S and sup S. Justify. n (8) Find the infimum, minimum, maximum and supremum of each of the following sets (write d.n.e. if it does not exist). S inf S min S max S sup S 1 1 - ² | neN} n {xER | x³-4x²+x+6 > 0 } = - = | m, n € N m n ∞ 1 2+ n n n=1 {XEQ x ≤ 2} (9) Show that the given collection F is an open cover for S such that it does not contain a finite subcover and so Sis not compact. - (a) S = (0, 2); and F = { Un | n E N} where Un. (-1,2-) (b) S = (0, ∞); and F = { Un | n E N} where Un = (0, n) 4 (10) Define the metric d on R² by d( (x₁, y₁), (x2, y2)) = max{ x₁ - x₂l|y₁ - yal}. Verify that this is a metric on R² and for > 0, draw an arbitrary e-neighborhood for a point (x, y) E R². Success never rests. On your worst days, be good; On your best days, be great, and on every other day, be better. Carmen Mariano ор of PUR DO
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