Greedy CART minimizes the listed split loss and terminates

At a node with finite nonempty sample set $I$, each coordinate has finitely many distinct observed values. Arrange them in increasing order $u_1<\cdots <u_r$. For every $k<r$, the midpoint between $u_k$ and $u_{k+1}$ gives two nonempty children. Conversely, any real threshold with two nonempty children has a unique gap $u_k\le s<u_{k+1}$ and induces the same partition as that midpoint. Thus the algorithm's candidate list is finite and contains every nonempty partition produced by one coordinate threshold.

For each candidate the child score is a finite real sum. If the candidate list is nonempty, repeatedly comparing its finite list of scores produces a minimum, and the coordinate/cut-index order selects a unique first minimizer among ties. By [F2], the regression child means attain the least squared error on each child; by [F1], the classification score is the declared weighted impurity. The chosen candidate therefore minimizes the declared child loss over all such partitions.

Every recursive call increases depth by one, and calls at depth $d_{\max }$ stop. Each split creates two disjoint nonempty sample sets from its parent, so every terminal leaf contains at least one of the $n$ sample indices. Each split increases the number of leaves by one; hence there are at most $n-1$ splits. The recursion terminates with depth at most $d_{\max }$ and all created children nonempty.