This seems to be false. Take the progression 0, 1/16, 2/16, 3/16, ..., p = 2 and q = 16. Then the second term is 1/16 = 1/q, 9th term is 8/16 = 1/p, but the 32th term is 31/16, not 1.
Books are not written by the great Authority. They are written by humans.
Prove it out. An arithmetic progression has a starting value (call it 'a') and a constant difference (call it 'd').
"the pth term is 1/q"
a + (p-1)d = 1/q
"the 9th term is 1/p"
a + (9-1)d = 1/p
This leads to
p*q is going to have to be an integer. You tell me how this can be and then the implications of the result.