12.1 Infinite Sets The following theorem follows directly from our previous work with the NIP and will be very handy later. It basically says that a sequence of nested closed intervals will still have a non-empty intersection even if their lengths do not converge to zero as in the NIP. Theorem 12.1.1. Let ([an,bn]), be a sequence of nested intervals such that lim bn – an> 0. Then there is at least onec E R such that c E [an, bn] for all z=1 n E N. Proof. By Corollary 10.4.5 of Chapter 10, we know that a bounded increasing sequence such as (an) converges, say to c. Since an < am < b, for m > n and lim am = c, then for any fixed n, a, 0. Show that there are at least two points, c and d, such that c E [an, bn] and d E [an, bn] for all n E N.
12.1 Infinite Sets The following theorem follows directly from our previous work with the NIP and will be very handy later. It basically says that a sequence of nested closed intervals will still have a non-empty intersection even if their lengths do not converge to zero as in the NIP. Theorem 12.1.1. Let ([an,bn]), be a sequence of nested intervals such that lim bn – an> 0. Then there is at least onec E R such that c E [an, bn] for all z=1 n E N. Proof. By Corollary 10.4.5 of Chapter 10, we know that a bounded increasing sequence such as (an) converges, say to c. Since an < am < b, for m > n and lim am = c, then for any fixed n, a, 0. Show that there are at least two points, c and d, such that c E [an, bn] and d E [an, bn] for all n E N.
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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![12.1 Infinite Sets
The following theorem follows directly from our previous work with the NIP and
will be very handy later. It basically says that a sequence of nested closed
intervals will still have a non-empty intersection even if their lengths do not
converge to zero as in the NIP.
Theorem 12.1.1. Let ([an, bn]), be a sequence of nested intervals such that
lim |bn
n=1
an > 0. Then there is at least one c E R such that c E [an, bn] for all
пEN.
Proof. By Corollary 10.4.5 of Chapter 10, we know that a bounded increasing
sequence such as (an) converges, say to c. Since an < am < bn for m > n and
= c, then for any fixed n, an <c< bn. This says c e [an, bn] for all n E N.
lim am
m→∞
Corollary 10.4.5. Let (xn) be a bounded, increasing sequence of real
numbers. That is, x1 < x2 < x3 <.…. Then (xn) converges to some real
number c.
in-context
Problem 12.1.2. Suppose lim |b,
0. Show that there are at least two
An
points, c and d, such that c E [an, bn] and d E [an, bn] for all n E N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0335de1d-d88b-4764-a43c-e2195c6bbbda%2F8d6b866e-06d1-4f23-b5ba-b364bd65753a%2Ft8a6slh_processed.png&w=3840&q=75)
Transcribed Image Text:12.1 Infinite Sets
The following theorem follows directly from our previous work with the NIP and
will be very handy later. It basically says that a sequence of nested closed
intervals will still have a non-empty intersection even if their lengths do not
converge to zero as in the NIP.
Theorem 12.1.1. Let ([an, bn]), be a sequence of nested intervals such that
lim |bn
n=1
an > 0. Then there is at least one c E R such that c E [an, bn] for all
пEN.
Proof. By Corollary 10.4.5 of Chapter 10, we know that a bounded increasing
sequence such as (an) converges, say to c. Since an < am < bn for m > n and
= c, then for any fixed n, an <c< bn. This says c e [an, bn] for all n E N.
lim am
m→∞
Corollary 10.4.5. Let (xn) be a bounded, increasing sequence of real
numbers. That is, x1 < x2 < x3 <.…. Then (xn) converges to some real
number c.
in-context
Problem 12.1.2. Suppose lim |b,
0. Show that there are at least two
An
points, c and d, such that c E [an, bn] and d E [an, bn] for all n E N.
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