If $g_1,g_2,g_3$ are each continuous functions and converge to a continuous function $g$. I am trying to show that for any test function $T$, we have that $\langle g_n, T\rangle$ converges to $\langle g,T \rangle$.
Here is what i have so far:
\begin{align*} \lim_{n\to\infty}\langle g_n, T \rangle &=\int_{-\infty}^\infty g_n(x)T(x)dx\\ &=\int_{-N}^N g_n(x)T(x)dx\\ &=\int_R\mathbf{1}_{[]-N,N]}g_n(x)T(x) \end{align*} I am not sure how to go from here (obviously I want the limit to somehow get inside the integral), I think here is where the continuity of the $(g_n)$ come in but I am unsure.