The question is:
Let $T$ be a continuous random variable with survivor function $S$ defined on the interval $[0, \omega]$.
Now consider the random variable $S(T)$, the survivor function evaluated at the unknown lifetime value $T$.
Show that $S(T)$ has a Uniform$[0,1]$ distribution.
My attempt at answering it is:
$P(S(T) \leq x) = P(T \leq S^{-1}(x))$
Where $S^{-1}(x) = \inf\left\lbrace t : S(t) \leq x \right\rbrace$
So then
$P(T \leq S^{-1}(x)) = 1 - P(T > S^{-1}(x)) = 1 -S(S^{-1}(x)) = 1 - x$
But the cdf of a Uniform$[0,1]$ distribution should be $x$ not $1-x$?