Bounded convergence theorem with filtrations

Question

Suppose in a filtered probability space $(\Omega,\mathbb{P},\{\mathcal{F}_n\},\mathcal{F})$ we have a sequence of almost surely vanishing adapted random variables $\varepsilon_n$. By the conditional bounded convergence theorem $\mathbb{E}_{\mathcal{G}}(e^{it\varepsilon_{n+1}}-1)\longrightarrow 0$ almost surely for any $\mathcal{G}\subset\mathcal{F}$. Does this implies also that $\mathbb{E}_{\mathcal{F}_n}(e^{it\varepsilon_{n+1}}-1)\longrightarrow 0$? I feel like it should, tried to prove it using $\mathcal{F}_\infty=\sigma\left(\bigcup_{n=0}^\infty\mathcal{F}_n\right)$, but could not get anywhere... Thanks in advance for any advice!