False
Convergent sequences are sequences that have a limit, meaning they approach a specific value as the sequence progresses. However, not all convergent sequences are bounded. Here's why:

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

Explanation:
A sequence is bounded if there exists a real number M such that |an| ≤ M for all n. While it is true that every convergent sequence is bounded in the sense that the terms of the sequence do not become arbitrarily large as n approaches infinity, it is not true that every convergent sequence is bounded in the sense that there exists a specific bound M that all terms of the sequence adhere to.

Counterexample:
Consider the sequence {an} = {n} for n ∈ ℕ. This sequence converges to infinity as n approaches infinity, but it is not bounded since there is no real number M such that |n| ≤ M for all n.
Therefore, it is false to say that every convergent sequence is bounded.