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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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TRANSCRIBE THE FOLLOWING TEXT IN DIGITAL FORMAT
![Solutio
We know that the unim of two compact sets
is compact.
(1)
Consider the statement
P(n): UK nehere each CK 1 l≤ken ip
a Compecct set.
·K=1
for n=1
P(1) : V₁ CK=C₁, nehich is compact as we assumed.
.: P(1) is true.
for 11=2
з ск
P(2): K=1
3 CK = C₁ UC₁₂, nehich is compact as it is a
Well known result stated in (1)
1
Let us assume that P.(n) is tome for some 12.
n
nt CK = (UCK) U (n+1 [using property of union]
Cnt
Now Pintl). U CK =
K=1
7 K=1
o
p(n) it true for some 17,2 =>
ut S= BCK i-e. S is compact.
вск
OCK is compact
Hence p(n+1): Wt CK = SUC₁+1
K=1
•Using (1) p(n+1) 1A true.
Hence by the principle of induction, pln) is true for all nEN.
prone of
that our result](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F860a1374-20cc-47e2-9366-3835646e97ec%2F445fe026-576a-401a-b38a-b6f1614ff415%2Fvqgu52r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solutio
We know that the unim of two compact sets
is compact.
(1)
Consider the statement
P(n): UK nehere each CK 1 l≤ken ip
a Compecct set.
·K=1
for n=1
P(1) : V₁ CK=C₁, nehich is compact as we assumed.
.: P(1) is true.
for 11=2
з ск
P(2): K=1
3 CK = C₁ UC₁₂, nehich is compact as it is a
Well known result stated in (1)
1
Let us assume that P.(n) is tome for some 12.
n
nt CK = (UCK) U (n+1 [using property of union]
Cnt
Now Pintl). U CK =
K=1
7 K=1
o
p(n) it true for some 17,2 =>
ut S= BCK i-e. S is compact.
вск
OCK is compact
Hence p(n+1): Wt CK = SUC₁+1
K=1
•Using (1) p(n+1) 1A true.
Hence by the principle of induction, pln) is true for all nEN.
prone of
that our result
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