Can we apply the Dominated convergence theorem of Lebesgue to the sequence of functions $ f_n(x) =\frac{n}{x^2 +n^2}, \quad x\in \mathbb{R}$ ?
It is $ f_n(x) =\frac{n}{x^2 +n^2} \leq \frac{n}{2nx} =\frac{1}{2x}$ but the function $1/x$ is not Lebesgue integrable. How can I find a function $ g \in L^+( \mathbb{R})$ such that $|f_n| \leq g, n\in \mathbb{N}$ almost everywhere in $\mathbb{R}$ so I can use Lebesgue's theorem?