Given a sequence \( \left(u_{n}\right)_{n \geq 1} \). Which one of the following statemients is true? A If \( \left(u_{n}\right)_{n \geq 1} \) is bounded then it converges B If \( \left(u_{n}\right)_{n \geq 1} \) converges then it is bounded C \( \left(u_{n}\right)_{n \geq 1} \) is bounded if and only if it convergs D If \( \left(u_{n}\right)_{n \geq 1} \) has a limit then it converges

Let's take a closer look at the statements provided: The second statement is true: If \( \left(u_{n}\right)_{n \geq 1} \) converges, then it is bounded. This is a fundamental result in analysis. When a sequence converges to a limit, all the terms in the sequence must eventually get arbitrarily close to that limit, meaning there exists some bounds on the sequence that keep it from going to infinity. However, the converse isn't necessarily true, as a bounded sequence doesn't always converge. As for the other statements, the first one is false because a bounded sequence may still oscillate and not converge. The third statement is also false for the same reason; a sequence can be bounded without converging. The fourth statement is misleading, as having a limit in the context defined is typically used interchangeably with convergence. Thus, the true statement here is indeed the one regarding convergence leading to boundedness.