已知命题 $p: \forall x \in R , a x^2+2 x+3>0$ 为真命题, 则实数 $a$ 的取值范围是 ( )
$\text{A.}$ $\left\{a \left\lvert\, 0 < a \leq \frac{1}{2}\right.\right\}$
$\text{B.}$ $\left\{a \left\lvert\, 0 < a < \frac{1}{3}\right.\right\}$
$\text{C.}$ $\left\{a \left\lvert\, a \geq \frac{1}{3}\right.\right\}$
$\text{D.}$ $\left\{a \left\lvert\, a>\frac{1}{3}\right.\right\}$
$\text{E.}$
$\text{F.}$