If $f_n \leq g$ where g is integrable and $f_n \to f$ then is it true that $\int f \leq \int g $?

Question

Hi I'm just curious as I prepare for a final, I cooked up this problem and wanted to know the answer. Suppose all $f_n>0$ and $f_n \leq g$ for all n and x with g integrable and $f_n \to f$. Then does that imply $\int f \leq \int g$?

I know by dominated convergence that $\int f_n \to \int f$ and $\int f_n \leq \int g_n$ for all n. But I don't know if the inequality holds in the limit as well. Thanks!

Answer 1

Yes: $f_n\leq g$ for all $n$ and $f_n\to f$ implies that $f\leq g$, hence $\int f\leq \int g$.

Answer 2

$$\int f = \lim_n \int f_n \leq \lim_n \int g = \int g$$