Find Uniform Minimum Variance Unbiased estimator (UMVU) using Lehmann Scheffé - showing statistic is complete

For some measurable function $g$, suppose

$$\mathbb E_{\theta}\left[g(S)\right]=\int_{\theta}^\infty g(x)ne^{-n(x-\theta)}\,dx=0\quad\,\forall\,\theta\in\mathbb R$$

That is, $$\int_{\theta}^\infty g(x)e^{-nx}\,dx=0\quad\forall\,\theta$$

Now for some $a\in(\theta,\infty)$, we can rewrite the last equation as

$$\int_{\theta}^a g(x)e^{-nx}\,dx+\int_a^\infty g(x)e^{-nx}\,dx=0\quad\forall\,\theta$$

Differentiating both sides of the last equation with respect to $\theta$, we get

$$g(\theta)e^{-n\theta}=0\quad\forall\,\theta$$

Now that $e^{-n\theta}>0$ for each $\theta$, you can conclude that $g$ is exactly zero almost everywhere.

This perhaps is a more convincing argument.