# Thread: Bounded and Convergence Proof Question

1. ## Bounded and Convergence Proof Question

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

Prove the Claim.

2. 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 $\displaystyle (a_{n})$ is a bounded sequence with the property that every convergent subsequence of $\displaystyle (a_{n})$ converges to the same limit a $\displaystyle \in \Re$, then $\displaystyle (a_{n})$ must converge.
Prove the lemma: that there cannot exist two disjoint closed intervals each containing infinitely many points of $\displaystyle a_n$. (Use the given and Bolzano-Weierstrauss)

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

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

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

Prove the Claim.
I'm probably making this more complicated than it has to be

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

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

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

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