Let $g(t) = E[X_{t}^{2}]$, the starting point is the following set of inequalities which hold for all $t \geq 0$: \begin{align} - \int _{0}^{t} 2g(s)ds + t \leq g(t) \leq \int _{0}^{t} 2g(s)ds + t \end{align}
Then, we need to prove the following:\begin{align} 0 < \text{lim inf}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}\leq \text{lim sup}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}} < \infty \end{align}
An obvious first step is integrating the above inequalities, and Fubini will probably also be useful, since then you can exchange the expectation and integrals. Furtermore, $X_{t}$ is also a submartingale, but I am not sure if that is relevant here.
However, even with the above knowledge I still can not picture where the limsup and liminf in the expression come from.
I would really appreciate any help.
EDIT: As per TheBridges' request, I also give the SDE of $X_{t}$ below, though I am not sure if it is relevant towards answering my question. \begin{align} dX_{t} = |X_{t}|dt + dW_{t}, X_{0} = 0 \end{align}
To derive the first inequalities at the top, I applied Ito to rewrite $X_{t}^2$, and together with the SDE of $X_{t}$ you can fairly easily derive these inequalities. But it is still unclear as to how they should be used to derive the second set of inequalities.