# Show that a sequence is bounded, monotone, using The Convergence Theorem

• MHB
• cbarker1
In summary, the conversation discusses the exercise from the book Basic Analysis by Jiri Lebel which asks to prove the sequence $(n+1)/n$ is monotone, bounded, and to use the monotone convergence theorem to find the limit. The conversation also mentions using the Archimedean Property to complete the proof and discusses finding the upper and lower bounds for the sequence.
cbarker1
Gold Member
MHB
Dear Every one,

In my book, Basic Analysis by Jiri Lebel, the exercise states
"show that the sequence $\left\{(n+1)/n\right\}$ is monotone, bounded, and use the monotone convergence theorem to find the limit"

My Work:
The Proof:
Bound
The sequence is bounded by 0.
$\left|{(n+1)/n}\right| \ge 0$
$n+1\ge0\, \forall\, n \in \Bbb{N}$

Monotone
$\frac{n+1}{n} \ge 0$$\frac{\left(n+1\right)+1}{n+1}\le\frac{n+1}{n} \frac{n+2}{n+1}\le\frac{n+1}{n} n^2+2n\le n^2+2n+1 the sequence is increasing monotone. Applying the theorem. \lim_{{n}\to{\infty}} \frac{n+1}{n}=\sup{{\frac{n+1}{n}:n\in\Bbb{N}}} Let A=\sup{{\frac{n+1}{n}:n\in\Bbb{N}}} We know the sup A is less than or equal to 1 as 1 is an upper bound. Take a number$$b\le1$$such that$$b\ge \frac{n+1}{n}$$.$$bn\ge n+1bn-n\ge1n(b-1)\ge1$$By the Archimedean Property..." What need I to do in order to finish the proof? THanks CBarker1 Cbarker1 said: the sequence is increasing monotone. I would say it is decreasing. Cbarker1 said: We know the sup A is less than or equal to 1 as 1 is an upper bound. Are you sure 1 is an upper bound? Cbarker1 said: Take a number$$b\le1$$such that$$b\ge \frac{n+1}{n}$$.$$bn\ge n+1bn-n\ge1n(b-1)\ge1$$By the Archimedean Property..." What need I to do in order to finish the proof? The sequence has a lower bound. Indeed, as you wrote, zero is a trivial lower bound. One is a slightly less trivial lower bound. Can you prove this? The sequence is also monotonously decreasing. So, you know it has a limit in$\mathbb{R}\$. Now you could try to show that the infimum of the sequence equals one, from which the value of the limit follows.

To finish the proof, you can use the fact that the sequence is bounded and monotone to apply the Monotone Convergence Theorem. This theorem states that if a sequence is bounded and monotone, then it must converge to a limit. In this case, since the sequence is increasing and bounded by 1, the limit must be 1. Therefore, you can conclude that the limit of the sequence is 1.

## 1. What is a sequence?

A sequence is an ordered list of numbers, usually denoted by {an} or (an). Each number in the sequence is called a term. For example, the sequence 1, 4, 7, 10, ... would be written as {1, 4, 7, 10, ...} or (1, 4, 7, 10, ...).

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

A sequence is bounded if the terms in the sequence are all between two fixed values, called the upper and lower bounds. In other words, the terms in the sequence do not get infinitely large or infinitely small as n increases. A bounded sequence can either be bounded above (all terms are less than or equal to a fixed value) or bounded below (all terms are greater than or equal to a fixed value).

## 3. How do you prove that a sequence is bounded?

To prove that a sequence is bounded, you need to show that the terms in the sequence are always between two fixed values. This can be done by finding the upper and lower bounds for the sequence and showing that all terms fall within those bounds. Another way to prove boundedness is by using the squeeze theorem, which states that if a sequence is bounded by two other sequences that converge to the same limit, then the original sequence must also converge to that limit.

## 4. What does it mean for a sequence to be monotone?

A sequence is monotone if the terms in the sequence either always increase or always decrease as n increases. A sequence can be monotone increasing (each term is greater than or equal to the previous term) or monotone decreasing (each term is less than or equal to the previous term).

## 5. How do you use the Convergence Theorem to show that a sequence is bounded and monotone?

The Convergence Theorem states that if a sequence is both bounded and monotone, then it must converge to a limit. To use this theorem to show that a sequence is bounded and monotone, you would first prove boundedness and monotonicity separately, and then use the Convergence Theorem to show that the sequence must converge to a limit. This limit would then be used to show that the sequence is both bounded and monotone.

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