Let $\{ \beta_n \}$ be a slowly increasing complex sequence. That is, there exists $M > 0$ and $k \in \mathbb{Z}$ such that $|\beta_n| \le C(1+ |n|)^k$ for all $n$ sufficiently large. Let $g_n$ be given by $$g_n(t) = \beta_n e^{2\pi i n t} $$ Show that $g_n$ converges to $0$ in $\mathcal{D}'(\mathbb{R})$ in the sense of distributions.
My attempt: $$|T_{g_n}(\phi) - 0| = |\int_\mathbb{R} \beta_n e^{2\pi i nt}\phi(t)dt| \le \int_\mathbb{R} |\beta_n e^{2\pi i nt}| |\phi(t)|dt \le ||\phi(t)||_\infty \int_\mathbb{R}|C(1+|n|)^ke^{2\pi i n t}|dt$$ $$\le ||\phi(t)||_\infty |C(1+|n|)^k| \int_\mathbb{R} |e^{2\pi i n t}|dt $$ But I'm not sure where to go from here, or if I've gone in the wrong direction from the beginning.