1, boundedness: A sequence or function is bounded. If its value range is within a certain range, it will not increase or decrease indefinitely. For example, a series of {1, -2, 3, -4, 5, ...} is bounded because its absolute value is always less than or equal to 5. In mathematics, boundedness is usually used to describe the behavior of a function or sequence.
2. Convergence: If the limit of a sequence or function exists and approaches a certain value, it is said that the sequence or function converges. For example, when n tends to infinity, the sequence {1/n} converges to 0. In mathematics, convergence is usually used to describe the extreme behavior of a sequence or function. The relationship between boundedness and convergence is that convergent sequences or functions are usually bounded, but bounded sequences or functions are not necessarily convergent. In other words, the convergent sequence or function must be bounded, but the bounded sequence or function does not necessarily converge. This is because even if a sequence or function has definite boundaries, its limit value does not necessarily exist.
Boundedness and convergence are two different concepts. Boundedness usually describes the range of values of a sequence or function, while convergence describes the limit behavior of a sequence or function. Bounded sequences or functions may converge, but not necessarily.