Show that sequence is unbounded

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In summary, the task is to show that the sequence an = sqrt(n) is unbounded. To do this, we use the contradiction method and assume that the sequence is bounded. However, by choosing n greater than M^2, where M is the supposed upper bound, we can find a term in the sequence that is greater than M, proving that the sequence is actually unbounded. This reasoning works for all real M.
  • #1
oceanthang
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Homework Statement


Show that the sequence an= sqrt(n) is unbounded


Homework Equations


there is no relevant equation require.


The Attempt at a Solution


Actually, I'm a newbie for real analysis, I try to prove it by using contradiction method,
but I stuck at half way, can anyone provide solution to me? thanks.
 
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  • #2
Suppose an is bounded. Then there exists M > 0 such that an <= M for all n. However this is absurd since you only have to pick n greater than M^2 to have an > M.

To illustrate: pick M = 120. 120^2 = 14400. a_14401 > 120.

Makes sense?
 
  • #3
Can you explain more detail about the illustration? I don't really get it about this statement
"only have to pick n greater than M^2 to have an > M", thanks.
 
  • #4
Could you show us what you did to become stuck half way? We can assist you by checking if your approach was going the right way.
 
  • #5
oceanthang said:
Can you explain more detail about the illustration? I don't really get it about this statement
"only have to pick n greater than M^2 to have an > M", thanks.

Stating that a_n is bounded means that you can find a value of M which is greater than all the terms in the sequence. You must prove there is no such M. Reduction to the absurd works well in this situation.

You suppose there exists M >= a_n for all n. But there is a problem with this statement, which is a_{[M^2] + 1} > M, where [M^2] is the integer part of M squared.

Suppose you state that M = 120.2 is greater than all the terms in the sequence (a purely arbitrary choice).
That is absurd since a_{[120.2^2]+1} = a_14449 = sqrt(14449) > 120.2. This reasoning works for all real M.
 
  • #6
I finally get it, thank you~
 

Related to Show that sequence is unbounded

1. What does it mean for a sequence to be unbounded?

When a sequence is unbounded, it means that there is no limit to how large or how small the terms in the sequence can get. In other words, the terms in the sequence can continue to grow or decrease infinitely.

2. How do you show that a sequence is unbounded?

To show that a sequence is unbounded, you need to prove that there is no number that can act as an upper or lower bound for the terms in the sequence. This can be done by showing that for any given number, there will always be a larger or smaller term in the sequence.

3. Can a sequence be both bounded and unbounded?

No, a sequence cannot be both bounded and unbounded. A sequence is either bounded, meaning there is a limit to how large or how small the terms can get, or unbounded, meaning there is no limit to how large or how small the terms can get.

4. What is the difference between a bounded and unbounded sequence?

The main difference between a bounded and unbounded sequence is that a bounded sequence has a limit to how large or small the terms can get, while an unbounded sequence does not have a limit and can continue to grow or decrease infinitely.

5. Why is it important to determine if a sequence is unbounded?

Determining if a sequence is unbounded is important because it helps us understand the behavior of the sequence. If a sequence is unbounded, it means that the terms can become arbitrarily large or small, which may have significant consequences in certain contexts, such as in mathematical proofs or in real-world applications.

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