Dominated Convergence Theorem: If $(f_n:n\geq 1)$ is a sequence of measurable functions and there exists a Lebesgue integrable function $g$ such that $|f_n|\leq g$ for all $n\geq 1$, then $\lim\int f_n\,d\mu=\int \lim f_n\,d\mu$.
My question: If we relax the hypothesis "$|f_n|\leq g$ for all $n\geq 1$" to "$|f_n|\leq g$ for all $n\geq N$", where $N$ is some positive integer, does the conclusion still hold?