(6) (10 %) Suppose that $f$ is Riemann integrable on $[0, A]$ for all $A < +\infty$, and $f(x) \to 1$ as $x \to +\infty$. Prove that

$$\lim{t\to 0} t \int0^{\infty} e^{-tx} f(x) \, dx = 1 \quad (t > 0).$$