# If a sequence is eventually bounded then it is bounded

• pixelate
In summary, the conversation discusses the concept of bounded sequences and a lemma that states if a sequence is eventually bounded, then it is bounded. The conversation also includes a question about a specific sequence, but it is determined that the sequence is not a valid example.
pixelate

## Homework Statement

Hi, I've been solving Calculus Deconstructed by Nitecki and I've been confused by a particular lemma in the book. Namely:

If a sequence is eventually bounded, then it is bounded:
that is, to show that a sequence is bounded, we need only find a number
γ ∈ R such that the inequality

$$|x_i| < \gamma \ \text{holds for all i ≥ K, for some K}$$

## Homework Equations

3. The Attempt at a Solution
[/B]
However if we consider the sequence ##\tan {\frac{\pi}{n}} ##, with n starting at 1. For terms when n is very high, we can find some value for ##\gamma## for which the condition holds and thus the sequence is bounded (according to the lemma). Yet at n=1, ##\tan {\frac{\pi}{n}} = \infty##, so the sequence cannot be bounded.

My questions are:

1. Is my reasoning at all correct in the first place ?

2. It is easy to see that my sequence does have a lower bound but no upper bound, so does the lemma only refer to one bound (But doesn't bounded mean bounded in both directions?)

pixelate said:

## Homework Statement

Hi, I've been solving Calculus Deconstructed by Nitecki and I've been confused by a particular lemma in the book. Namely:

If a sequence is eventually bounded, then it is bounded:
that is, to show that a sequence is bounded, we need only find a number
γ ∈ R such that the inequality

$$|x_i| < \gamma \ \text{holds for all i ≥ K, for some K}$$

## Homework Equations

3. The Attempt at a Solution
[/B]
However if we consider the sequence ##\tan {\frac{\pi}{n}} ##, with n starting at 1. For terms when n is very high, we can find some value for ##\gamma## for which the condition holds and thus the sequence is bounded (according to the lemma). Yet at n=1, ##\tan {\frac{\pi}{n}} = \infty##, so the sequence cannot be bounded.

My questions are:

1. Is my reasoning at all correct in the first place ?

2. It is easy to see that my sequence does have a lower bound but no upper bound, so does the lemma only refer to one bound (But doesn't bounded mean bounded in both directions?)

You don't have a valid sequence, so that is not a counterexample to anything.

Anyway, at ##n = 1## we have ##\tan(\pi) = 0##, so that is OK. The trouble occurs at ##n = 2##.

But, I say it again: you don't have a valid sequence of real numbers with this example.

Ray Vickson said:
But, I say it again: you don't have a valid sequence of real numbers with this example.

.Oh I see, that's very illuminating thank you. Of course at n=2, the the element of the sequence is undefined and it is not infinite at all. I feel silly for making such a basic mistake now.

## 1. What does it mean for a sequence to be eventually bounded?

For a sequence to be eventually bounded means that after a certain point in the sequence, all of the terms are contained within a specific range or interval. This range can be either finite or infinite, but the main idea is that the sequence eventually becomes bounded.

## 2. How is eventual boundedness different from regular boundedness?

Regular boundedness means that every term in the sequence is contained within a specific range or interval. This range can be either finite or infinite. In eventual boundedness, only after a certain point in the sequence, all of the terms are contained within a specific range or interval.

## 3. What is the significance of a sequence being eventually bounded?

The significance of a sequence being eventually bounded is that it allows us to make certain conclusions about the behavior of the sequence. For example, we can make statements about the limit or convergence of the sequence based on its eventual boundedness.

## 4. Can a sequence be eventually bounded but not bounded?

No, a sequence cannot be eventually bounded but not bounded. If a sequence is eventually bounded, then it is also bounded. This is because eventual boundedness means that after a certain point in the sequence, all of the terms are contained within a specific range or interval, which is the definition of boundedness.

## 5. How can we prove that a sequence is eventually bounded?

To prove that a sequence is eventually bounded, we need to find a specific value or term in the sequence after which all of the terms are contained within a specific range or interval. This can be done by using the definition of boundedness and finding a suitable upper and lower bound for the sequence after a certain point.

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