Let $F:\Bbb{R}[0,1]$ be a cumulative probability distribution function. We define the generalised inverse (quantile function) by $$F^{-1}(u)=inf\{x:F(x)\geq u\}.$$

Show $$F(F^{-1}(F(x)))=F(x),$$ and conclude that the distribution function $F$ assigns no mass to the interval $(F^{-1}(F(x)),x]$.

Any help on this would be really appreciated - I have managed to show $F(F^{-1}(u))\geq u$ and $F^{-1}(F(x))\leq x$, and am unsure if this would help to prove $F^{-1}(u)=inf\{x:F(x)\geq u\}$