Multidimensional Bolzano-Weierstrass Theorem. If $(X_n)_{n \in \mathbb{N}} \subset \mathbb{R}^n$ is a bounded sequence, then $(X_n)$ contains a convergent subsequence.
Let $X_n = ((X_n)_1, \dots, (X_n)_n)$ be the sequence vector. Since the sequence vector is bounded iff every coordinate is bounded, we know $(X_n)_1$ is bounded. Applying one-dimensional B-W, we get a strictly increasing $k_1: \mathbb{N} \to \mathbb{N}$ such that $(X_{n_{k_1}})_1$ is a convergent subsequence ($k_1$ being the indexing function).
Then we move on to the subsequence $(X_{n_{k_1}})_2$ which is also bounded. B-W implies there's another $k_2: \mathbb{N} \to \mathbb{N}$ such that $(X_{n_{{k_1}_{k_2}}})_2$ is a convergent subsequence.
Repeating this process until the last coordinate, we get
$$(X_{n_{k^*}})_n$$
where $k^* = k_n \circ k_{n-1} \circ \dots \circ k_1$ is a composition of strictly increasing functions (itself strictly increasing). Then
$$X'_n = ((X_{n_{k^*}})_1, \dots, (X_{n_{k^*}})_n)$$
is a convergent subsequence of the original sequence because: (i) the last coordinate is convergent by construction, (ii) the others are subsequences of convergent subsequences, therefore, they themselves converge.
Is this appeal to indexing functions correct?