Dominated Convergence Theorem (for sequences)

The Dominated Convergence Theorem states that if a sequence of functions converges point-by-point to a limit function, and all these functions are bounded by a single, integrable function, then the limit of their integrals equals the integral of their limit function. Essentially, under these conditions, we can exchange the limit operation and integration, ensuring that as the functions get closer to the limit, their integrals also approach the integral of that limit. This theorem is crucial for dealing with limits inside integrals in analysis and probability theory.