Q:

Prove lim as as $n \to \infty \int_{[-n,n]}{f} = \int_{\mathbb{R}}f$

Prove that $$\lim_{n \to \infty} \int_{[-n,n]}{f} = \int_{\mathbb{R}}f.$$

We're given $f$ a nonnegative measurable function on $\mathbb{R}$.
So far I have:
Let $f_n = 1_{[-n,n]}f$ then $\{f_n\}$ is nonnegative and monotone and $ f \to f_n$ pointwise.
By MCT, $$\lim_{n \to \infty}  \int_{[-n,n]}{f} = \lim_{n \to \infty}  \int_{[-n,n]}{f_n} = \lim_{n \to \infty}  \int_{\mathbb{R}}{f} = \int_{\mathbb{R}}{f}.$$
Is this right?  I'm a little concerned about my last line

A:

What you are doing is basically correct, but you are not writing it properly. 
You have that $f_n\nearrow f$ (it is essential that the convergence is monotone). 
Then
$$
\lim_n\int_{[-n,n]}f=\lim_n\int_{\mathbb R} f_n=\int_{\mathbb R}\lim_n f_n
=\int_{\mathbb R}f,
$$
where the Monotone Convergence Theorem is used in the second equality. 

