\left|\int_{-\infty}^{+\infty}\delta(x)\eta_n(x)-\int_a^b\delta(x)\eta_n(x)\,dx\right|
=\left|\int_{-\infty}^{a}\delta(x)\eta_n(x)+\int_b^{+\infty}\delta(x)\eta_n(x)\,dx\right|\leq\int_{a-1/n}^a|\delta(x)\eta_n(x)|\,dx+\int_b^{b+1/n}|\delta(x)\eta_n(x)|\,dx\leq
\int_{a-1/n}^a|\delta(x)|\,dx+\int_b^{b+1/n}|\delta(x)|\,dx