= Solution
Let $Q=F^{-1}$ be the <quantile function>. For $p\in(0,1)$,
$$
Q(p)\leq t\quad\Longleftrightarrow\quad p\leq F(t).
$$
The reverse implication is immediate from the infimum defining $Q$. For the forward implication, if $Q(p)<t$ there is a point $s\leq t$ with $F(s)\geq p$. If $Q(p)=t$, choose $s_m<t+1/m$ with $F(s_m)\geq p$; monotonicity and right continuity give
$$
F(t)=\lim_{m\to\infty}F(t+1/m)\geq p.
$$
The endpoint values $p=0,1$ have zero probability under a continuous uniform variable.
Consequently
$$
\mathbb P(Q(U)\leq t)
=\mathbb P(U\leq F(t))
=F(t).
$$
Thus the <inverse transform sampling> identity is
$$
\boxed{F^{-1}(U)\sim F}.
$$
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