= Solution
For $0<j_1<\cdots<j_n<t$, the probability of one jump in each infinitesimal interval around $j_1,\ldots,j_n$ and no other jump by time $t$ gives the joint density
$$
\lambda^n e^{-\lambda t}
$$
for $(J_1,ldots,J_n)$ together with the event $N_t=n$. Since
$$
\mathbb P(N_t=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!},
$$
the conditional density is
$$
\frac{\lambda^ne^{-\lambda t}}
{e^{-\lambda t}(\lambda t)^n/n!}
=\frac{n!}{t^n}
$$
on that ordered simplex.
Now $n$ independent <uniform distribution>[uniform] points on $[0,t]$ have joint density $t^{-n}$ on the cube. Each ordered vector has $n!$ permutations, so their <order statistics> have density $n!/t^n$ on the same simplex. The densities agree, proving <Poisson process conditional arrival times>.
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