= Solution
Expand an initial bipartite <pure state> in local <orthonormal bases> as
$$
|\psi\rangle=\sum_{i,j}C_{ij}|i\rangle_A|j\rangle_B.
$$
By the <Schmidt decomposition>, or equivalently the <singular value decomposition> of $C$, its <Schmidt rank> is $\operatorname{rank}C$. In a specified local measurement branch, the product <Kraus operator> $A\otimes B$ changes the coefficient <matrix> to
$$
C'=ACB^{\mathsf T}.
$$
Indeed the coefficient of $|k\rangle|l\rangle$ is $\sum_{i,j}A_{ki}C_{ij}B_{lj}$. The elementary <matrix rank> inequality gives
$$
\operatorname{rank}(ACB^{\mathsf T})\leq\operatorname{rank}C.
$$
Normalizing a nonzero branch multiplies $C'$ by a scalar and therefore leaves its <matrix rank> unchanged.
Now refine a complete <LOCC> transcript to record every individual <Kraus operator>. Although later choices depend on the earlier classical outcomes, fixing a transcript fixes all those choices. Composing the local operations therefore gives a single product <Kraus operator> $A_\tau\otimes B_\tau$ in that branch. Each nonzero branch obeys the same <Schmidt-rank contraction under product operators>. Local auxiliary systems prepared independently are included by local isometries and do not add <entanglement>. Discarding a local auxiliary system is included by resolving its <partial trace> in an <orthonormal basis>; this again yields refined product <Kraus operators>.
If the protocol produces a specified <pure state> deterministically, its output <density operator> is a sum of positive rank-one branch <density operators>. Every nonzero branch vector must be proportional to that output vector: its squared overlap with any vector orthogonal to the output sums to zero, so each such overlap is separately zero. Thus the output <Schmidt rank> equals that of each surviving branch, and
$$
\boxed{r_{\mathrm{out}}\leq r_{\mathrm{in}}.}
$$
This proves <monotonicity of Schmidt rank under LOCC>, even for a nonzero branch selected by <postselection>. If the classical transcript is instead discarded and the result is mixed, the refined branch vectors supply an ensemble with <Schmidt ranks> at most $r_{\mathrm{in}}$; by definition its <Schmidt number> is at most $r_{\mathrm{in}}$. This is the correct mixed-state extension, rather than a bound on the ranks of its <reduced density matrices>. In finite dimensions, any limiting exact protocol obeys the same pure-state bound because the set of coefficient <matrices> of rank at most $r_{\mathrm{in}}$ is closed, being defined by vanishing minors.
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