Bounded Monotonic Sequence Theorem

In summary: So, in summary, to use the Bounded Monotonic Sequence Theorem, the sequence must be bounded and monotonic. In this case, the sequence is bounded by a lower bound of 0 and is monotonic increasing, therefore it is convergent.
  • #1
Euler2718
90
3

Homework Statement


[/B]
Use the Bounded Monotonic Sequence Theorem to prove that the sequence:

[tex] \{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\} [/tex]

Is convergent.

Homework Equations

The Attempt at a Solution


[/B]
I've shown that it has an upper bound and is monotonic increasing, however it is to my understanding that for me to use this theorem the sequence must be bounded (and of course have monotonicity) - i.e. have an upper and lower bound. I can't seem to show that this has a lower bound; even graphically, it just blows off to negative infinity. How should I proceed?
 
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  • #2
Morgan Chafe said:

Homework Statement


[/B]
Use the Bounded Monotonic Sequence Theorem to prove that the sequence:

[tex] \{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\} [/tex]

Is convergent.

Homework Equations

The Attempt at a Solution


[/B]
I've shown that it has an upper bound and is monotonic increasing, however it is to my understanding that for me to use this theorem the sequence must be bounded (and of course have monotonicity) - i.e. have an upper and lower bound. I can't seem to show that this has a lower bound; even graphically, it just blows off to negative infinity.
Why do you think this? You didn't say anything about the possible values of i, but I assume they are positive integers {1, 2, 3, ...}.

Consider ##f(x) = x - \sqrt{x^2 + 1}##. All of the values of this function are negative, since ##x < \sqrt{x^2 + 1}## for all x > 0. As x gets larger, the difference is smaller, and approaches zero. The largest difference comes when x is smallest (i.e., closest to zero).
Morgan Chafe said:
How should I proceed?
 
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  • #3
Morgan Chafe said:

Homework Statement


[/B]
Use the Bounded Monotonic Sequence Theorem to prove that the sequence:

[tex] \{a_{i} \} = \Big\{ i - \sqrt{i^{2}+1} \Big\} [/tex]

Is convergent.

Homework Equations

The Attempt at a Solution


[/B]
I've shown that it has an upper bound and is monotonic increasing, however it is to my understanding that for me to use this theorem the sequence must be bounded (and of course have monotonicity) - i.e. have an upper and lower bound. I can't seem to show that this has a lower bound; even graphically, it just blows off to negative infinity. How should I proceed?

If ##f(i) = i -\sqrt{i^2+1}## is monotonically increasing, then ##f(1) < f(2) < f(3) < \cdot##, so if ##f(1)## is a finite number, it is a lower bound!
 
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  • #4
Mark44 said:
Why do you think this? You didn't say anything about the possible values of i, but I assume they are positive integers {1, 2, 3, ...}.

Consider ##f(x) = x - \sqrt{x^2 + 1}##. All of the values of this function are negative, since ##x < \sqrt{x^2 + 1}## for all x > 0. As x gets larger, the difference is smaller, and approaches zero. The largest difference comes when x is smallest (i.e., closest to zero).
Ray Vickson said:
If ##f(i) = i -\sqrt{i^2+1}## is monotonically increasing, then ##f(1) < f(2) < f(3) < \cdot##, so if ##f(1)## is a finite number, it is a lower bound!
Thank you, I think I got it now.
 

Related to Bounded Monotonic Sequence Theorem

What is the Bounded Monotonic Sequence Theorem?

The Bounded Monotonic Sequence Theorem, also known as the Monotone Convergence Theorem, states that a bounded monotonic sequence, meaning a sequence that is either increasing or decreasing and has an upper and lower bound, will converge to a limit.

How is the Bounded Monotonic Sequence Theorem used in mathematics?

The Bounded Monotonic Sequence Theorem is an important tool in mathematical analysis and is used to prove the convergence of sequences in various mathematical concepts, such as calculus and real analysis.

Can you provide an example of a bounded monotonic sequence?

Sure, one example of a bounded monotonic sequence is an = 1/n. This sequence is bounded between 0 and 1, and is also monotonically decreasing as each term is smaller than the previous one. This sequence converges to a limit of 0.

What is the significance of the Bounded Monotonic Sequence Theorem?

The Bounded Monotonic Sequence Theorem is significant because it provides a powerful tool for mathematicians to prove the convergence of sequences, which is crucial in many mathematical proofs and applications.

Is the Bounded Monotonic Sequence Theorem always true?

Yes, the Bounded Monotonic Sequence Theorem is always true for any bounded monotonic sequence. However, it is important to note that the theorem only applies to sequences in real analysis and may not hold true in other mathematical contexts.

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