Limit of Sequence: Proving $b_0\in L(a_n)$ and Convergence Condition

In summary, the conversation discusses sequences and limit points. It is proposed that a sequence converges to a limit in $\mathbb{R}$ if and only if each subsequence has a subsequence that converges to that limit. The correctness of this proposition is then questioned and demonstrated through a proof by contradiction. The conversation also delves into the definition of limit points and how they relate to sequences.
  • #1
mathmari
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Hey! :eek:

Let $(a_n)_{n=1}^{\infty}$ be a real sequence and let $(b_n)_{n=1}^{\infty}$ a sequence in the set of limit points of $(a_n)_{n=1}^{\infty}$, $L(a_n)$.

There is also a $b_0\in \mathbb{R}$ with $b_n\rightarrow b_0$ for $n\rightarrow \infty$.

I want to show that then $b_0\in L(a_n)$.

How could we show this? (Wondering) I want to show also that a sequence $(a_n)_{n=1}^{\infty}$ converges to $a\in \mathbb{R}$ iff each subsequence $(a_{n_k})_{k=1}^{\infty}$ of $(a_n)_{n=1}^{\infty}$ has a subsequence $(a_{n_{k_j}})_{j=1}^{\infty}$ that converges to $a$.

We have that a sequence converges to a limit in $\mathbb{R}$ iff each subsequence converges to that limit, or not?
But how could we show the above condition? (Wondering)
 
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  • #2
For the second question is it maybe as follows? (Wondering)

Proposition:
A sequence converges to a limit in $\mathbb{R}$ iff each subsequence converges to that limit$\Leftarrow$ :
We suppose that each subsequence $(a_{n_k})_{k=1}^{\infty}$ of $(a_n)_{n=1}^{\infty}$ has a subsequence $(a_{n_{k_j}})_{j=1}^{\infty}$ that converges to $a$.
From the proposition we get that each subsequence has to converge to $a$.
Again from the proposition we get that the sequence has to converge to $a$.

$\Rightarrow$ :
We suppose that not each subsequence $(a_{n_k})_{k=1}^{\infty}$ of $(a_n)_{n=1}^{\infty}$ has a subsequence $(a_{n_{k_j}})_{j=1}^{\infty}$ that converges to $a$. Then there are some subseuqneces that vonverge to an other point, say $b\neq a$. So, it cannit be that the sequnec converges to $a$. Is this correct? (Wondering)
 
  • #3
For the second question I changed it.. $\Leftarrow$ :
We suppose that $(a_n)$ does not converfe to $a$, so there is at least two limit points, so there are two different subsequences that converge to different points, say $a$ and $b$.
From the proposition we have that the subsequence that converges to $b$ has no subsequence that converges to $a$.

$\Rightarrow$ :
We suppose that $(a_n)$ converges to $a$. From the proposition we have that every subsequence converges to $a$. If we apply the proposition to each subsequence we get that each subsequence of esch subsequence converges to $a$. Is this correct? (Wondering)
 
  • #4
mathmari said:
Let $(a_n)_{n=1}^{\infty}$ be a real sequence and let $(b_n)_{n=1}^{\infty}$ a sequence in the set of limit points of $(a_n)_{n=1}^{\infty}$, $L(a_n)$.

There is also a $b_0\in \mathbb{R}$ with $b_n\rightarrow b_0$ for $n\rightarrow \infty$.

I want to show that then $b_0\in L(a_n)$.

How could we show this? (Wondering)
We have that $b_n\in H(a_n)$ is a limit point of $(a_n)$ if every neighbourhood of $b_n$ contains at least one point of $(a_n)$ different from $b_n$ itself, right? (Wondering)

Let $y_n$ be that point.
So we have then that $|y_n-b_n|>0$ ? (Wondering)

If $b_n=b_0$ for some $n$ then we have that $b_0\in L(a_n)$.
If $b_n\neq b_0, \forall n$ then from the above defintion $0<|y_n-b_n|<|b_n-b_0|$.

Is this correct? Does this help? (Wondering)
 
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Related to Limit of Sequence: Proving $b_0\in L(a_n)$ and Convergence Condition

1. What is the definition of a limit of sequence?

The limit of a sequence is the value that a sequence approaches as the number of terms increases towards infinity. It is denoted by limn→∞an and can be thought of as the "end behavior" of a sequence.

2. What is the relationship between the limit of a sequence and its terms?

If a sequence has a limit, then its terms will get closer and closer to that limit as the number of terms increases. In other words, the terms of a sequence will "converge" to the limit value.

3. How do you prove that a term is in the limit of a sequence?

To prove that a term, b0, is in the limit of a sequence, an, you must show that for any positive number ε, there exists a positive integer N such that |b0 - an| < ε for all n > N. This means that b0 is within ε distance from the limit value for all terms after a certain index N.

4. What is the convergence condition for a sequence?

The convergence condition for a sequence is the requirement that the sequence has a limit. In other words, the sequence must approach a specific value as the number of terms increases towards infinity.

5. How do you determine if a sequence satisfies the convergence condition?

To determine if a sequence satisfies the convergence condition, you can use various convergence tests such as the limit comparison test, ratio test, or root test. These tests evaluate the behavior of the terms in the sequence and determine if they approach a limit or tend towards infinity.

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