Optional time plus constant is a Stopping time

Question

If $T$ is an optional time and $\theta$ a constant >0, I want to show, that $T+\theta$ is a stopping time regarding the standard filtration $(\mathcal{F}_t)_{t\in\mathbb{R}_+}$. I have already shown, that $\{T+\theta<t\}\in\mathcal{F}_t$, However since it is not given, that $(\mathcal{F}_t)_{t\in\mathbb{R}_+}$ is right continuous, I do not know how to proceed. Any hints would be appreciated.