# Compare the limit of a sequence with the upper bound of the sequence as in Dominated convergence theorem

Recall Dominated Convergence Theorem: Let $$(f_n)_{n\geq 1}$$ be a sequence of measurable functions s.th. $$f_n\rightarrow f$$ pointwise. Suppose there exists a non-negative integrable function $$g$$ with $$|f_n|\leq g$$ for all $$n\geq 1$$. Then $$$$\lim_{n\rightarrow \infty}\int_Sf_nd\mu=\int_S fd\mu.$$$$

My question is that: Is it true $$f\leq g$$?

Yes, of course. For all $$x$$ and $$n$$, $$f_n(x)\le \lvert f_n(x)\rvert\le g(x)$$ Therefore, for all $$x$$, $$f(x)=\limsup_{n\to\infty} f_n(x)\le g(x)$$