Nested intervals 2.22. Prove that to every set of nested intervals [a,, b,], n = 1, 2, 3, . .. there corresponds one and only one real number. By definition of nested intervals, ans 1 2 an, bn41, < b,n = 1, 2, 3, ... and lim (a,- b,) = 0. Then a, < a, < bn s b, and the sequences {a,} and {b,} are bounded and, respectively, monotonic increasing and decreasing sequences and so converge to a and b. To show that a = b and thus prove the required result, we note that b-a = (b- b,) + (b,– an) + (a, – a) (1) |b-al < [b-b,| + |b, - a,| + la,-a| (2) Now, given any e > 0, we can find N such that for all n>N |b-b,| < e/3, |b, -a|
Nested intervals 2.22. Prove that to every set of nested intervals [a,, b,], n = 1, 2, 3, . .. there corresponds one and only one real number. By definition of nested intervals, ans 1 2 an, bn41, < b,n = 1, 2, 3, ... and lim (a,- b,) = 0. Then a, < a, < bn s b, and the sequences {a,} and {b,} are bounded and, respectively, monotonic increasing and decreasing sequences and so converge to a and b. To show that a = b and thus prove the required result, we note that b-a = (b- b,) + (b,– an) + (a, – a) (1) |b-al < [b-b,| + |b, - a,| + la,-a| (2) Now, given any e > 0, we can find N such that for all n>N |b-b,| < e/3, |b, -a|
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2.22) My professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.
![Nested intervals
2.22.
Prove that to every set of nested intervals [a, b,], n = 1, 2, 3, . .. there corresponds one and only one real
number.
By definition of nested intervals, a 1 2 a, bas1, < b,n= 1, 2, 3, . .. and lim (a, - b,) = 0.
Then a, < a, < b, < b, and the sequences {a,} and {b„} are bounded and, respectively, monotonic
increasing and decreasing sequences and so converge to a and b.
To show that a = b and thus prove the required result, we note that
b-a = (b – b,) + (b, – an) + (a, – a)
(1)
|b-al < lb-b,| + |b, - a,| + la, - a|
(2)
Now, given any e > 0, we can find N such that for all n> N
|b-b,| < €/3, |b, - al <e/3,
(3)
so that from Equation (2), |b - al <e. Since e is any positive number, we must have b – a = 0 or a = b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad530511-9758-42d9-9994-894bc4eea3f3%2Fbae21f3f-d2b0-4aeb-8fb8-03856df871df%2Fll67aj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Nested intervals
2.22.
Prove that to every set of nested intervals [a, b,], n = 1, 2, 3, . .. there corresponds one and only one real
number.
By definition of nested intervals, a 1 2 a, bas1, < b,n= 1, 2, 3, . .. and lim (a, - b,) = 0.
Then a, < a, < b, < b, and the sequences {a,} and {b„} are bounded and, respectively, monotonic
increasing and decreasing sequences and so converge to a and b.
To show that a = b and thus prove the required result, we note that
b-a = (b – b,) + (b, – an) + (a, – a)
(1)
|b-al < lb-b,| + |b, - a,| + la, - a|
(2)
Now, given any e > 0, we can find N such that for all n> N
|b-b,| < €/3, |b, - al <e/3,
(3)
so that from Equation (2), |b - al <e. Since e is any positive number, we must have b – a = 0 or a = b.
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