Inequalities of the quantile function I'm trying to rigorously prove the following inequalities involving the quantile function $Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\}$ where $F$ is the distribution function:
1) $F(x) < a \iff x < Y(a)$
2) $a \leq F(x) \iff  Y(a) \leq  x$
Any help would be appreciated. Thanks.
 A: As pointed by @cardinal related answers are available at Quantile function properties.
For any $p\in (0,1)$ and $F$ cumulative distribution function of a given random variable, let us define the quantile function $Q$:
$$Q(p) \triangleq \inf \{x\in \mathbb{R} : p\leq F(x) \}$$
This is also called the generalized inverse distribution function, and it is denoted by $F^{-1}$. Note first that $F$ is monotone increasing and does not, in general, have an inverse (hence the generalized terminology above). 
Note also that $Q$ is monotone increasing too (based on the fact that, if $A\subseteq B$ and $\inf A$, $\inf B$ exist, then $\inf A \geq \inf B$).
Moreover, $Q(F(x))\leq x$ for all $x$ and $F(Q(p))\geq p$ for all $p$, simply by the definition of $\inf$ and $Q$ itself.
Using these statements, we get that $Q(p)\leq x$ if and only if $p\leq F(x)$.
Hope it helps. (Please don't use $\omega$ as a variable of the cdf or quantile function. It's usually reserved for elements in the sample space $\Omega$ on which the random variable is defined, hidden here.)
