About Monotone Convergence Theorem

Question

I have the following problem:

Let $(f_n)_{n\geq1}$ a sequence of integrable functions such that $f_n\colon\mathbb{R}\to\mathbb{R}$ and $$\sup_{n\in\mathbb{N}}\int_{\mathbb{R}}f_n(x)dx<\infty.$$ If $f_{n+1}(x)\geq f_{n}(x)$ for all $n\in\mathbb{N}$ and $x\in\mathbb{R}$, and $f(x)=\lim_{n\to\infty}f_n(x)$, show that $\displaystyle \int_{\mathbb{R}}fdx<\infty$ and $$\lim_{n\to\infty}\int_{\mathbb{R}}f_n(x)dx=\int_{\mathbb{R}}f(x)dx.$$

It seems that I have to use the Monotone Convergence Theorem, but I think that nothing ensures the non-negativity of the $f_n$'s.

If anyone has some hint, it would be appreciated.

Answer 1

If you define the sequence of functions $$ g_n(x) := f_n(x) - f_1(x),$$ then $(g_n)_{n \geq 1}$ is both increasing and positive.

Apply Monotone Convergence to $(g_n)_{n \geq 1}$.

Answer 2

By your assumption, $(\int_\mathbb{R} f_n)$ is a uniformly bounded sequence, so the monotone convergence theorem tells us that $\int_\mathbb{R} f<\infty.$ To make a positive sequence, why don't you just consider $g_n=f-f_n?$ Since the $f_n$'s are increasing, you know that $g_n\geq 0,$ and $g_n\nearrow 0.$ Now, we see that $$\int_\mathbb{R} f-\int_\mathbb{R} f_n=\int\limits_\mathbb{R} g_n\rightarrow 0$$ (you can also see the integrability of $f$ from here).