= Solution
The <monotone convergence theorem> says that if $0\leq f_j\uparrow f$ almost everywhere on a measure space, then $\int f_j\uparrow\int f$, allowing infinity. The <dominated convergence theorem> says that if $g_j\to g$ almost everywhere and $|g_j|\leq H$ for an integrable $H$, then $g$ is integrable and $\int g_j\to\int g$.
For $a\ne0$, the truncated functions $f_a\mathbf1_{[-j,j]}$ increase to $f_a$. Each has the elementary integral $2\arctan(|a|j)/|a|$. Thus monotone convergence gives
$$
\boxed{\int_{\mathbb R}f_a(x)\,dx=\frac\pi{|a|}<\infty.}
$$
For fixed $x$, eventually $|x|\leq n\pi/2$ and $n\sin(x/n)\to x$, so $g_n(x)\to1/(1+x^2)$. Inside the support the supplied inequality yields $n^2\sin^2(x/n)\geq4x^2/\pi^2$. Outside the support $g_n=0$. Consequently the global bound is
$$
0\leq g_n(x)\leq\frac1{1+4x^2/\pi^2},
$$
and this majorant is integrable by the already proved formula with $a=2/\pi$. Dominated convergence now gives $\boxed{\int g_n\to\int(1+x^2)^{-1}dx=\pi}$. The majorant is independent of $n$, including on the moving support.
Back to article page
