Prove that the following statement is true whenever 1≤r<n, or else, find a counterexample.

Given a 2n x n matrix of 0's and 1's with the column sum and the row sum respectively equal to 2r and r, it is always possible to mark 2n of the 1's such that:
Precisely one 1 is marked in each row, and exactly two 1's are marked in each column.

*** Adapted from PME 479 by Herbert Taylor.