In an effort to use the Bounded Convergence Theorem, I'd like to show that for a fixed value $M$ that $f_n(x) = \frac{n}{1+n^2x^2} < M$. Taking the derivative, it's clear that 0 is the greatest value of $f_n$ in this interval, but that leaves me with $f_n < n$, where $n$ is not fixed. Any pointers? Can I just fix $x \neq 0$? Thanks in advance.
EDIT: the domain of integration is $[0,1].$
\begin{align*} \int_{0}^{1}\dfrac{n}{1+n^{2}x^{2}}dx&=\int_{0}^{n}\dfrac{1}{1+x^{2}}dx\\ &=\tan^{-1}n\\ &\rightarrow\dfrac{\pi}{2}. \end{align*} Bounded Convergence Theorem does not work.