It turns out that there is a direct relationship between Cantor's lemma for nested compact sets $K_n$, on the one hand, and nested sequences of internal sets, on the other. Namely, the latter can be viewed as a special case of the former as follows.

The decreasing nested sequence of internal sets, $\langle{}^\ast\!K_n\colon n\in\mathbb N\rangle$, has a common point $x$ by saturation. But for a compact set $K_n$, every point of ${}^\ast\!K_n$ is nearstandard (i.e., infinitely close to a point of $K_n$). In particular, $st(x)\in K_n$ for all $n$, as required.

It turns out that there is a direct relationship between Cantor's lemma for nested compact sets $K_n$, on the one hand, and nested sequences of internal sets, on the other. Namely, the latter can be viewed as a special case of the former as follows.

The decreasing nested sequence of internal sets, $\langle{}^\ast\!K_n\colon n\in\mathbb N\rangle$, has a common point $x$ by saturation. But for a compact set $K_n$, every point of ${}^\ast\!K_n$ is nearstandard, i.e., infinitely close to a point of $K_n$ (in other words, every point is in the halo of a standard point of $K_n$). In particular, $st(x)\in K_n$ for all $n$, as required.