Assume that $X_n$ is a sequence of a zero-mean and unit variance random variables (and maybe having density w.r.t. to Lebesgue). Can we conclude that $ P(X_n \in [0,R_n]) $ is bounded away from zero eventually, say $$\liminf_{n \to \infty} P(X_n \in [0,R_n]) > 0$$ assuming that $R_n \to \infty$. Intuitively, $X_n$ should put some nonvanishing amount of mass on the positive real line because of the zero-mean assumption and the condition of unit variance should prevent that mass from escaping to infinity (?)

Here is a partial argument: We have \begin{align*} P(X_n \in [0,R_n]) &= 1 - P(X_n > R_n) - P(X_n < 0) \\ &\ge 1 - P(|X_n| > R_n) - P(X_n < 0) \\ &\ge 1 - \frac{E X_n^2}{R_n^2} - P(X_n < 0) \\ &= 1 - o(1) - P(X_n < 0) \end{align*} Thus, the problem reduces to arguing that $P(X_n < 0)$ stays away from 1.