Proof that every convergent sequence is bounded
Every convergent sequence is bounded because a convergent sequence approaches a limit as the number of terms in the sequence increases. This implies that eventually, all terms of the sequence will be contained within a certain neighborhood of the limit.

Definition of a convergent sequence
A sequence {an} is said to converge to a limit L if for every ε > 0, there exists an N such that for all n ≥ N, |an - L| < />

Definition of a bounded sequence
A sequence {an} is said to be bounded if there exists a real number M such that |an| ≤ M for all n.

Proof that a convergent sequence is bounded
1. Let {an} be a convergent sequence that converges to a limit L.
2. By the definition of convergence, for ε = 1, there exists an N such that for all n ≥ N, |an - L| < />
3. Let M = max{|a1|, |a2|, ..., |aN-1|, |L| + 1}.
4. Then, for all n ≥ N, we have:
- |an| = |an - L + L| ≤ |an - L| + |L| < 1="" +="" |l|="" ≤="" />
5. Therefore, the sequence {an} is bounded by M.
6. Hence, every convergent sequence is bounded.

Conclusion
In conclusion, a convergent sequence is bounded because it approaches a limit, which allows us to find a bound for all its terms beyond a certain point in the sequence. This property is essential in the study of sequences and their convergence properties.