Final answer:
The sequence in question, with a unique fixed point and being bounded, will have a limit, meaning it will converge rather than diverge, oscillate, or increase indefinitely.
Step-by-step explanation:
If there is a unique fixed point in a fixed-point iteration, and the sequence is bounded, we can conclude that the sequence has a limit. This is based on the properties of convergence in bounded sequences. If a fixed-point iteration converges, it means that as you compute more terms in the sequence, they will get closer and closer to the fixed point.
A sequence that is bounded means that there is a value which the terms of the sequence will never exceed. The unique fixed point acts as an attractor for the values of the sequence, pulling successive iterations towards it. Therefore, rather than diverging, oscillating indefinitely, or constantly increasing, the bounded nature and the presence of a unique fixed point imply convergence to a limit. In other words, the sequence will get closer and closer to the fixed point with each iteration, ultimately stabilizing at that point provided the method is properly applied and the initial conditions allow for it.