# Proving Existence of Limit of Sequence {xn}

• cristina89
In summary, the limit of a sequence is the value that the terms of the sequence approach as the number of terms increases. To prove the existence of a limit, the terms of the sequence must approach a specific value. This can be proven through methods such as the epsilon-delta definition or the squeeze theorem. The epsilon-delta definition states that for any positive number epsilon, there exists a positive integer N such that for all terms with an index greater than N, the distance between the term and the limit is less than epsilon. A sequence can only have one limit, and the limit can be determined by analyzing the behavior of the terms as the number of terms increases. This can be done through finding a pattern, using formulas, or applying the
cristina89
Be {xn} a sequence that satisfies the condition 0 ≤ $x_{m+n}$ ≤ $x_{m}$ + $x_{n}$. Prove that $lim_{n ->∞}$ xn/n exists.

I'm kind of lost in this.

cristina89 said:
Be {xn} a sequence that satisfies the condition 0 ≤ $x_{m+n}$ ≤ $x_{m}$ + $x_{n}$. Prove that $lim_{n ->∞}$ xn/n exists.

I'm kind of lost in this.

I would start by thinking like this:

x2<=x1+x1

x3<=x2+x1<=x1+x1+x1

etc. What can you make of that?

Ughh I'm still lost :S

Hi cristina89!

Can you write xn in the form that Dick suggested?

## 1. What is a limit of a sequence?

The limit of a sequence refers to the value that a sequence of numbers approaches as the number of terms in the sequence increases. It is the value that the terms of the sequence get closer and closer to, but may never actually reach.

## 2. How do you prove the existence of a limit of a sequence?

To prove the existence of a limit of a sequence, one must show that the terms of the sequence approach a specific value as the number of terms increases. This can be done through various methods such as the epsilon-delta definition or the squeeze theorem.

## 3. What is the epsilon-delta definition of a limit?

The epsilon-delta definition of a limit states that for any given positive number epsilon, there exists a positive integer N such that for all terms of the sequence with an index greater than N, the distance between the term and the limit is less than epsilon.

## 4. Can a sequence have more than one limit?

No, a sequence can only have one limit. If a sequence has more than one limit, then it is not a convergent sequence and does not have a limit.

## 5. How do you determine the limit of a sequence?

The limit of a sequence can be determined by analyzing the behavior of the terms of the sequence as the number of terms increases. This can be done through various methods such as finding a pattern, using mathematical formulas, or applying theorems such as the squeeze theorem.

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