I am studying for my Measure Theory test, and we are working on Convergence (Monotone, Dominant, and Bounded) in order to evaluate integrals using the result that $\int f=\lim_{n\to\infty}\int f_n=\int\lim_{n\to\infty}f_n$. I understand how to do problems using monotone convergence, but I am having difficulty with using the other two convergence theorems.
For example, one practice problem is to use dominant convergence to evaluate $\lim_{n\to\infty}\int_0^1(\frac{nx}{1+n^2x^2})dx$.
The theorem can be used if you use $f_n$, a sequence of functions, that is Riemann Integrable, $\vert f_n\vert\leq g$ for some $g\in L^1$, and that $f_n\to f$ almost everywhere.
I have that $f_n=\frac{nx}{1+n^2x^2}\chi_{[0, 1]}$, and since $f_n$ is a continuous function on a finite interval, $f_n\in L^1$. Also, $\vert f_n\vert\leq\frac{1}{x^2}$, and $\frac{1}{x^2}\in L^1$.
From here I do not know exactly what to do about showing that $f_n\to f$ almost everywhere, but I know that the answer to $\lim_{n\to\infty}\int_0^1(\frac{nx}{1+n^2x^2})dx$ is 0 just from knowing how to integrate, and the fact that the theorem is suppose to hold true for this problem.
The function $f(x) = \frac 1{x^2}$ does not belong to $L^1([0,1])$.
On the other hand, the function $\phi(t) = \frac{t}{1 + t^2}$, $t \ge 0$, attains its maximum value $\frac 12$ at $t = 1$.
Thus if $x \in [0,1]$ and $n \ge 0$ then $$|f_n(x)| = f_n(x) = \frac{nx}{1 + (nx)^2} \le \frac 12$$ so that $g(x) = \frac 12$ is a suitable majorant for the sequence. You can apply LDCT to conclude $$\lim_{n \to \infty} \int_0^1 f_n = \int_0^1 \lim_{n \to \infty} f_n = 0.$$