Define the sine integral $\operatorname{Si}(x)$ by$$\def\Si{\operatorname{Si}}\Si(x) = \int^x_0 \frac{\sin t}{t} \, dt$$
I want to establish upper and lower bounds on $\frac{1}{\pi} \Si(\pi x)$. The answer here proves a closely-related upper bound. But heuristically, in my particular case, tighter upper and lower bounds appear to be given by$$\frac{1}{2} - \frac{1}{3 \pi x} \leq \frac{1}{\pi} \Si(\pi x) \leq \frac{1}{2} + \frac{1}{3 \pi x}$$
Mathematica suggests that the inequalities hold out at least as far as $x = 1{,}000{,}000{,}000$, and plots look reassuring:
Are these bounds valid, and if so can they be proved?