To prove \( CM = DN \) based on the given information about the cube \( ABCD \) and points \( M \) and \( N \) located on edges \( AB \) and \( BC \), respectively, you can follow these steps:

1. **Understand the Geometry**: Since \( ABCD \) is a cube, all edges are equal in length, and opposite edges are parallel and equal.

2. **Identify the Segments**:
- Let \( AB = a \), where \( a \) is the length of the edge of the cube.
- If \( M \) is on \( AB \) and \( N \) is on \( BC \), you can assign coordinates:
- \( M \) can be represented as \( (x_M, 0, 0) \) where \( 0 \leq x_M \leq a \).
- \( N \) can be represented as \( (a, y_N, 0) \) where \( 0 \leq y_N \leq a \).

3. **Calculate Distances**:
- The distance \( CM \) can be calculated as follows:
\[
CM = \sqrt{(x_M - a)^2 + (0 - 0)^2 + (0 - 0)^2} = a - x_M
\]
- The distance \( DN \):
\[
DN = \sqrt{(a - a)^2 + (y_N - a)^2 + (0 - 0)^2} = a - y_N
\]

4. **Equality Condition**:
- The statement \( CM = DN \) implies that:
\[
a - x_M = a - y_N \implies x_M = y_N
\]
- This means that as long as \( M \) is placed on edge \( AB \) and \( N \) on edge \( BC \) such that their respective distances from point \( A \) are the same, \( CM \) will equal \( DN \).

5. **Conclusion**:
\( CM = DN \) holds true under the condition that the distances from point \( A \) to points \( M \) and \( N \) are equal, confirming the assertion in the question when they are accordingly positioned.

This proof leverages the symmetry and properties of the cube, showing that the distances from point \( C \) to point \( M \) and from point \( D \) to point \( N \) can indeed be equal.