 Solution

For a continuous strictly increasing cumulative distribution function $F$ and a uniform distribution $U$ on $(0,1)$, monotonicity givesThus each transformed draw has distribution function $F$. This is inverse transform sampling. The conclusion extends to discontinuous or non-strictly increasing $F$ by its quantile function $Q(u)=\inf \{x:F(x)\ge u\}$: right continuity gives $Q(u)\le y$ exactly when $u\le F(y)$. Independent uniforms produce independent transformed draws, but marginal uniformity alone does not establish independence.

If “uniformly distributed sequence” instead means deterministic equidistribution, the empirical fraction of ${\eta }_i\le y$ is the empirical fraction of ${\xi }_i\le F(y)$ and tends to $F(y)$. That is an empirical distribution statement, not a claim that the deterministic sequence consists of independent random variables.