Sequence has convergent subsequence

In summary, the conversation discusses proving the existence of a convergent subsequence in a sequence $(b_n)$, given that $(a_n)$ and $(c_n)$ are bounded and $a_n \leq b_n \leq c_n$. One approach is using the Bolzano-Weierstrass theorem, but there may be other ways to show it.
  • #1
evinda
Gold Member
MHB
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Hello! (Wave)

Let $(a_n), (b_n), (c_n)$ sequences such that $(a_n), (c_n)$ are bounded and $a_n \leq b_n \leq c_n$ for each $n=1,2, \dots$ I want to show that $(b_n)$ has a convergent subsequence.

I have thought the following:

Since $(a_n), (c_n)$ are bounded, $\exists m_1, m_2 \in \mathbb{Z}$ such that $m_1 \leq a_n$ and $c_n \leq m_2$.

Then $m_1 \leq b_n \leq m_2$, i.e. $(b_n)$ is bounded. So, from Bolzano-Weierstrass theorem, $(b_n)$ has a convergent subsequence.

But... is the above complete? Could we also show it somehow else? (Thinking)
 
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  • #2
Hey evinda,

Looks good to me.
 
  • #3
GJA said:
Hey evinda,

Looks good to me.

Nice, thank you... (Smirk)
 

1. What does it mean for a sequence to have a convergent subsequence?

When a sequence has a convergent subsequence, it means that there is a subset of the sequence that approaches a specific limit or value as the number of terms in the subset increases.

2. How is a convergent subsequence different from a convergent sequence?

A convergent sequence is a sequence in which all the terms approach a specific limit or value. A convergent subsequence, on the other hand, is a subset of a sequence that approaches a specific limit or value as the number of terms in the subset increases, while the rest of the sequence may not converge.

3. Can a sequence have more than one convergent subsequence?

Yes, a sequence can have multiple convergent subsequences. This occurs when there are multiple subsets of the sequence that approach different limits or values as the number of terms in the subset increases.

4. How can we identify a convergent subsequence in a sequence?

To identify a convergent subsequence in a sequence, we need to find a subset of the sequence that approaches a specific limit or value as the number of terms in the subset increases. This can be done by examining the terms in the sequence and looking for any patterns or trends.

5. What is the importance of a sequence having a convergent subsequence?

A sequence having a convergent subsequence is important because it tells us that there are certain subsets of the sequence that have a clear limit or value. This can help us understand the behavior of the sequence as a whole and make predictions about its future terms.

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