I used a similar idea but not exactly induction.
I defined a function F(k) = (number of black beads seen in the last m+n beads) - m, where k varies between m+n to 2m+2n.
If F(m+n) == 0, we are done, since we can cut the necklace at the midpoint.
If not, let us assume that F(m+n) > 0. In which case, F(2m+2n) must necessarily be < 0, since the second half of the necklace must have less than m black beads. Now we have a function whose value goes from positive to negative, and at every step the window shifts by exactly one bead. This means that there must exist a point p somewhere between m+n and 2m+2n, where F(p) = 0. We can guarantee the existence of such a point because, at every intermediate point, we drop one bead from the window at the left, and add one bead to the window at the right, as we keep moving the window from m+n to 2m+2n.
A similar argument can be applied to the case F(m+n) < 0.
