after proving an is bounded can you use the Bolzano weistrass theorum to prove that an is convergent also

No, proving that a sequence is **bounded** alone is not enough to guarantee that it is **convergent**. The Bolzano-Weierstrass Theorem states that every **bounded** sequence has a **convergent subsequence**, but this does not imply that the whole sequence is convergent. Here's a breakdown:

### Steps:
1. **Boundedness**: A sequence \(\{a_n\}\) is bounded if there exists some real number \(M\) such that \(|a_n| \leq M\) for all \(n\). Boundedness alone doesn't guarantee convergence, but it restricts the values of the sequence to a specific range.

2. **Bolzano-Weierstrass Theorem**: This theorem tells us that if a sequence is **bounded**, there exists at least one **subsequence** that conver

Explore whether proving a sequence is bounded is enough to guarantee convergence. Understand the role of the Bolzano-Weierstrass Theorem, which states that every bounded sequence has a convergent subsequence, but this does not imply the whole sequence is convergent. Suitable for undergraduate students studying real analysis.

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