S 1 {1- | neN} n inf S min S max S sup S

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Ans. no. 8 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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