Bounded and Convergence Proof Question

**Janu42** · March 11th 2010, 07:45 AM

I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

Claim: If $(a_{n})$ is a bounded sequence with the property that every convergent subsequence of $(a_{n})$ converges to the same limit a $\in \Re$ , then $(a_{n})$ must converge.

Prove the Claim.

**Plato** · March 11th 2010, 08:48 AM

Originally Posted by Janu42

assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.
Claim: If $(a_{n})$ is a bounded sequence with the property that every convergent subsequence of $(a_{n})$ converges to the same limit a $\in \Re$ , then $(a_{n})$ must converge.

Prove the lemma: that there cannot exist two disjoint closed intervals each containing infinitely many points of $a_n$ . (Use the given and Bolzano-Weierstrauss)

By Bolzano-Weierstrauss the set $\left\{a_n:n\in \mathbb{N}\right\}$ has a limit point $L$ .

Use the lemma to prove: if $\varepsilon > 0$ then the interval $\left[ {L - \varepsilon ,L + \varepsilon } \right]$ contains almost all the terms of $a_n$ .

**southprkfan1** · March 11th 2010, 08:57 AM

Originally Posted by Janu42

I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

Claim: If $(a_{n})$ is a bounded sequence with the property that every convergent subsequence of $(a_{n})$ converges to the same limit a $\in \Re$ , then $(a_{n})$ must converge.

Prove the Claim.

I'm probably making this more complicated than it has to be

Proof by contradiction

Suppose $(a_{n})$ does not converge to a. Then, there exists an e>0 and a sequence of points $(b_{n})$ such that
l $b_{n}$ - al > e for all n=1,2,3....

Look at the sequence $(b_{n})$ . By Bolzano-Weierstrauss, it has a convergent subsequence $(c_{n})$ , which is also a subsequence of $(a_{n})$ . So $(c_{n})$ must converge to a. But l $c_{n}$ - al > e for all n...contradiction.

**harish21** · March 11th 2010, 10:00 AM

Originally Posted by Janu42

I didn't get this on our first exam, and I'm trying to figure it out. I'm assuming Bolzano-Weierstrauss is used in this at some point, but I can't figure out how to formally prove the statement.

Claim: If $(a_{n})$ is a bounded sequence with the property that every convergent subsequence of $(a_{n})$ converges to the same limit a $\in \Re$ , then $(a_{n})$ must converge.

Prove the Claim.

I had posted a similar question, and got this response.

check this thread out:

http://www.mathhelpforum.com/math-he...bsequence.html

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