Let $A_t:[0,\infty)\to[0,\infty]$ be a non-decreasing, possibly infinite, right-continuous function. Define $C_s:=\inf\{t:A_t>s\}$. It is clear that $C$ is non-decreasing and therefore its left and right limits exist, specifically $C_{s-}:=\lim_{u\nearrow s}C_u$ is well defined for every $s\geq0$.
This is the setting from Revuz and Yor's book "Continuous Martingales and Brownian Motion" when dicussing generalized inverses (chapter zero, section 4). They say "it is easily seen that $C_{s-}= \inf\{t:A_t\geq s\}$."
I've tried multiple different ways to properly justify this, but have come up against issues every time.
How can one rigourously justify this claim in this setting?