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Unbounded sequence

  • #1
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Homework Statement


Either give an example or show that no example exists.

An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound.

Homework Equations




The Attempt at a Solution



example: {k}3k=-∞

and by this notation i mean that k starts at -∞ and ends at 3, and k∈ℤ .

I chose -∞ as the lower bound because I wanted this sequence to be unbounded. Then I chose 3 as the upper bound because that is what the original statement asked for.


My questions:
Does this example fulfill the original statement because I am unsure of what the question means by "and no β < 3 is an upper bound." The book uses β as the upper bound in previous pages.

Also does the way I wrote the sequence mean what I want it to mean? Because I am also confused on the notation.
 

Answers and Replies

  • #2
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Homework Statement


Either give an example or show that no example exists.

An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound.
How can an unbounded sequence have an upper bound?
fishturtle1 said:

Homework Equations




The Attempt at a Solution



example: {k}3k=-∞
Can you list the first, say, five terms in this sequence?
fishturtle1 said:
and by this notation i mean that k starts at -∞ and ends at 3, and k∈ℤ .

I chose -∞ as the lower bound because I wanted this sequence to be unbounded. Then I chose 3 as the upper bound because that is what the original statement asked for.


My questions:
Does this example fulfill the original statement because I am unsure of what the question means by "and no β < 3 is an upper bound." The book uses β as the upper bound in previous pages.

Also does the way I wrote the sequence mean what I want it to mean? Because I am also confused on the notation.
 
  • #3
Ray Vickson
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Homework Statement


Either give an example or show that no example exists.

An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound.

Homework Equations




The Attempt at a Solution



example: {k}3k=-∞

and by this notation i mean that k starts at -∞ and ends at 3, and k∈ℤ .

I chose -∞ as the lower bound because I wanted this sequence to be unbounded. Then I chose 3 as the upper bound because that is what the original statement asked for.


My questions:
Does this example fulfill the original statement because I am unsure of what the question means by "and no β < 3 is an upper bound." The book uses β as the upper bound in previous pages.

Also does the way I wrote the sequence mean what I want it to mean? Because I am also confused on the notation.
Perhaps an "unbounded sequence" means what everybody else calls an "infinite sequence". An infinite sequence can be bounded or unbounded.
 
  • #4
LCKurtz
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Homework Statement


Either give an example or show that no example exists.

An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound.

Homework Equations




The Attempt at a Solution



example: {k}3k=-∞

and by this notation i mean that k starts at -∞ and ends at 3, and k∈ℤ .

I chose -∞ as the lower bound because I wanted this sequence to be unbounded. Then I chose 3 as the upper bound because that is what the original statement asked for.


My questions:
Does this example fulfill the original statement because I am unsure of what the question means by "and no β < 3 is an upper bound." The book uses β as the upper bound in previous pages.

Also does the way I wrote the sequence mean what I want it to mean? Because I am also confused on the notation.
If you mean what I think you do, a better way to write it would be ##a_k = 4-k,~k = 1 ..\infty##.
 
  • #5
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This is the definition of an unbounded sequence I've been using from online,

a sequence is bounded if it is bounded above and below <=> if ∃k∈ℝ such that | xn | ≤ k ∀n∈ℕ.


The first five terms in this sequence would be
-∞, -∞+1, -∞+2, -∞+3, -∞+4


I was thinking that an unbounded sequence can have an upper bound if it goes to infinity in some direction but converges to a number as well.
 
  • #6
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28
I checked the answer in the back of the book, and the answer is fn=3-n which seems similar to ak=4-k, k=1...∞.

I think I would have gotten this same answer had the original statement been " An unbounded sequence for which 3 is an upper bound ".

I'm still confused by the bold part: "An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound. "
I think that it means no number less than 3 can be an upper bound, but isn't this redundant since we said 3 is an upper bound in the first part of the statement?
 
  • #7
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This is the definition of an unbounded sequence I've been using from online,

a sequence is bounded if it is bounded above and below <=> if ∃k∈ℝ such that | xn | ≤ k ∀n∈ℕ.


The first five terms in this sequence would be
-∞, -∞+1, -∞+2, -∞+3, -∞+4
No, these aren't numbers.
fishturtle1 said:
I was thinking that an unbounded sequence can have an upper bound if it goes to infinity in some direction but converges to a number as well.
 
  • #8
LCKurtz
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I checked the answer in the back of the book, and the answer is fn=3-n which seems similar to ak=4-k, k=1...∞.
They are the same, assuming ##n## starts at ##0## in ##3-n##.
I think I would have gotten this same answer had the original statement been " An unbounded sequence for which 3 is an upper bound ".

I'm still confused by the bold part: "An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound. "
I think that it means no number less than 3 can be an upper bound, but isn't this redundant since we said 3 is an upper bound in the first part of the statement?
No, it isn't redundant. Since ##3## is a term of the sequence, no number ##x## less than ##3## can be an upper bound. How could it be if ##x < 3##?
 
  • #9
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I checked the answer in the back of the book, and the answer is fn=3-n which seems similar to ak=4-k, k=1...∞.
This sequence is bounded since all of the terms are less than or equal to 3.
fishturtle1 said:
I think I would have gotten this same answer had the original statement been " An unbounded sequence for which 3 is an upper bound ".

I'm still confused by the bold part: "An unbounded sequence for which 3 is an upper bound, and no β < 3 is an upper bound. "
I think that it means no number less than 3 can be an upper bound, but isn't this redundant since we said 3 is an upper bound in the first part of the statement?
I think the problem is poorly worded.
 
  • #10
LCKurtz
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This sequence is bounded since all of the terms are less than or equal to 3.
I think the problem is poorly worded.
Aren't you confusing "bounded above" with "bounded"?
 
  • #11
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Aren't you confusing "bounded above" with "bounded"?
I don't think so. The definition of bounded sequence I am using is that it is a sequence that is bounded above and bounded below. That is, for some M > 0, |an| < M for all n in Z+ (or similar restriction on n).
See https://proofwiki.org/wiki/Definition:Bounded_Sequence
 
  • #12
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They are the same, assuming ##n## starts at ##0## in ##3-n##.


No, it isn't redundant. Since ##3## is a term of the sequence, no number ##x## less than ##3## can be an upper bound. How could it be if ##x < 3##?
Ok I think I get it. So we're told 3 is an upper bound. That means that the greatest number in this sequence is less than or equal to 3.

we're also told that no number less than 3 is an upper bound.

Therefore 3 must be included in this sequence.

If we were not told that no number less than 3 is an upper bound, then our sequence could have been something like xn=-10-n, n=0 . . ∞

Am i understanding this correctly?
 
  • #13
LCKurtz
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I don't think so. The definition of bounded sequence I am using is that it is a sequence that is bounded above and bounded below. That is, for some M > 0, |an| < M for all n in Z+ (or similar restriction on n).
See https://proofwiki.org/wiki/Definition:Bounded_Sequence
Then why do you say in post #9 that ##a_k = 4-k,~k=1..\infty## is bounded?
 
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  • #14
LCKurtz
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Ok I think I get it. So we're told 3 is an upper bound. That means that the greatest number in this sequence is less than or equal to 3.

we're also told that no number less than 3 is an upper bound.

Therefore 3 must be included in this sequence.

If we were not told that no number less than 3 is an upper bound, then our sequence could have been something like xn=-10-n, n=0 . . ∞

Am i understanding this correctly?
Yes, almost. You could have it true if the sequence just got arbitrarily close to ##3## but less than ##3##.
 
  • #15
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Ok, thank you both for your help, helped me a lot on this question .
 
  • #16
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Then why do you say in post #9 that ##a_k = 4-k,~k=1..\infty## is bounded?
I should have said "bounded above."
 
  • #17
LCKurtz
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I should have said "bounded above."
Right. That's what I pointed out in post #10 in the first place.
 

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