A function $q: (0,1) \rightarrow R $ is said to be a quantile function of $X$ if
$P[X<q(u)]\le u \le P[X\le q(u)]$ for all $u\in(0,1)$
The lower quantile function of $X$ is given by
$q_X^-(u).=inf\{x\in R:F_X(x)\leq u\}$
I want to show that it is a quantile function.
Would it be correct to state (first equality due to left-continuity of lower quantile function and second inequality from the definition):
$P[X<q_X^-(u)]= u = P[X\le q_X^-(u)]$ for all $u\in(0,1)$ to have the proof?