Question

How would you prove that N-S pole of a magnet cannot be separated

Answer 1

because like charges repel each other and unlike charges attract each other.
N-S pole are unlike charges

Answer 2

The magnetic effect is due to current loops which produce what is called a dipole field. Magnets exist only in the form of dipoles with two ends, a north and a south. When a bar magnet is split into two pieces, you don't get a separate north part and a south part