# Show the following sequence as a monotone increasing

• MHB
• cbarker1
In summary, the conversation discusses a sequence in a given set that is monotone increasing. The terms of the sequence are defined as ${x}_{n}$ and $S\ne\emptyset$, and it is given that ${x}_{n-1}<{x}_{n}\le\sup S$ for all $n\ge2$. The speaker is unsure of how to prove the sequence is monotone increasing, but another person reassures them that it is already given since the terms of the sequence satisfy the definition of strict monotonicity.
cbarker1
Gold Member
MHB
Dear Everyone, Here is the sequence: Let $S\subset\Bbb{R}$ and ${x}_{n}\in S$ and $S\ne\emptyset$ . ${x}_{n-1}<{x}_{n}\le\sup S$ for all $n\ge2$. Prove the sequence is monotone increasing.

I need help proving it; I do not know where to start? Thanks
Carter

Last edited:
I'm confused: Isn't this already given since $x_{n-1} < x_n$ for all $n \ge 2$, assuming $x_1$ is the first term in the sequence?

The assumption is correct where ${x}_{1}$ is given. So I can say the sequence is monotone increasing?

Yes. I mean, you are given terms $x_n$ in a set $S$ such that $x_{n-1} < x_n$ for all $n \ge 2$.
That latter inequality is the definition of strict monotonicity, so if the exercise really reads like this, then I cannot see what you would have to do else. (The supremum does not play any role, either.)

Anyone else here on board that sees something I overlooked?

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

A monotone increasing sequence is one in which each term is larger than or equal to the previous term. In other words, the sequence either stays the same or increases in value as you move along the sequence.

## 2. How can I tell if a sequence is monotone increasing?

To determine if a sequence is monotone increasing, you can compare each term to the previous term. If each term is larger than or equal to the previous term, then the sequence is monotone increasing. If there is at least one term that is smaller than the previous term, then the sequence is not monotone increasing.

## 3. Can a sequence be both monotone increasing and decreasing?

No, a sequence cannot be both monotone increasing and decreasing. A sequence is either monotone increasing or monotone decreasing, but not both at the same time.

## 4. What is the difference between a monotone increasing and monotone decreasing sequence?

A monotone increasing sequence increases in value as you move along the sequence, while a monotone decreasing sequence decreases in value as you move along the sequence. In other words, a monotone increasing sequence goes from smaller values to larger values, while a monotone decreasing sequence goes from larger values to smaller values.

## 5. How can I show a sequence as monotone increasing?

To show a sequence as monotone increasing, you can list out the terms in the sequence and compare each term to the previous term. If each term is larger than or equal to the previous term, then the sequence is monotone increasing. You can also graph the sequence and look for a consistent increase in values as you move along the graph.

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