Table of Contents

## Is monotone increasing bounded?

Only monotonic sequences can technically be called “bounded” The smallest value of an increasing monotonic sequence will be its first term, where n = 1 n=1 n=1. In this case, a n ≥ a 1 a_n\ge{a_1} an≥a1, so we know that increasing monotonic sequences are bounded below.

**When a monotonic decreasing sequence is convergent?**

Every monotonically increasing sequence which is bounded above is convergent. 3.1. 3 Theorem: If is monotonically decreasing and is bounded below, it is convergent.

**Are increasing sequences divergent?**

Any unbounded increasing sequence necessarily goes off to ∞, so perhaps the only nontrivial direction here is showing that every increasing, divergent sequence is unbounded. The contrapositive is to show that every bounded increasing sequence is convergent.

### How do you tell if a sequence is bounded or unbounded?

A sequence an is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded above because 1/n≤1 for all positive integers n. It is also bounded below because 1/n≥0 for all positive integers n.

**Can a strictly increasing sequence be bounded?**

The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly. A bounded monotonic increasing sequence is convergent. We will prove that the sequence converges to its least upper bound (whose existence is guaranteed by the Completeness axiom).

**Can a sequence be bounded by infinity?**

Each decreasing sequence (an) is bounded above by a1. We say a sequence tends to infinity if its terms eventually exceed any number we choose. Definition A sequence (an) tends to infinity if, for every C > 0, there exists a natural number N such that an > C for all n>N.

## Is every decreasing sequence bounded?

It is important to remember that any number that is always less than or equal to all the sequence terms can be a lower bound. Some are better than others however. A quick limit will also tell us that this sequence converges with a limit of 1.

**Can an increasing function be convergent?**

If {an} is bounded above and increasing then it converges and likewise if {an} is bounded below and decreasing then it converges.

**Can a sequence be convergent and not bounded?**

Answer The sequence {an = (−a)n} is bounded below by −1 and bounded above by 1, and so is bounded. This sequence does not converge, though; since |an − an+1| = 2 for all n, this sequence fails the Cauchy criterion, and hence diverges. For the other part, we know that every convergent sequence is bounded.

### How do you prove a sequence is bounded?

A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.

**Which is true of every bounded monotonic sequence?**

Thus, every bounded monotonic sequence is convergent. is increasing and bounded. that is, an + 1 > an for any natural number n, therefore the sequence is increasing.

**Is there a limit to a monotonic decreasing sequence?**

To show that it does indeed have a limit, we’ll prove that it is monotonic decreasing and bounded below. Since the terms of the sequence are positive, the sequence is clearly bounded below by 0.

## How is the limit value of a bounded sequence determined?

Bounded sequences, Monotonic sequence, Every bounded monotonic sequence is convergent. Limits of sequences. Properties of convergent sequences. The limit value is exclusively determined by the behavior of the terms in its close neighborhood. Bounded sequences.

**Is the infimum of a monotone sequence finite?**

If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit. Proof. Theorem. If { a n } {\\displaystyle \\{a_{n}\\}} is a monotone sequence of real numbers (i.e., if a n ≤ a n+1 for every n ≥ 1 or a n ≥ a n+1 for every n ≥ 1), then this sequence has a finite limit if and only if the sequence is bounded.