Let $k$ be an algebraically closed field of characteristic $0$. Let $T_n$ be the set of all possible log canonical threshold of a pair $(X,Y)$ where $X/k$ is a smooth variety and $Y \subseteq X$ is a nonzero closed subschemes. The following two facts are first provedproved (Tommaso de Fernex, Mircea Mustata, Limits of log canonical thresholds) via non-standard methods:
$T_n$ is closed in $\mathbb R$ for all $n$.
The set of points of accumulations from above of $T_n$ is $T_{n-1}$.
I think proofs that avoid non-standard analysis emergedemerged later (J. Kollár, Which powers of holomorphic functions are integrable?), but the first one used non-standard technique.