Prove (k,k+1) = 1 for all k element of Z

Let (k, k+1) = d, then d|k and d| k+1

so there exist n, m element of Z s.t. k =dn and k+1 = dm

Now, k +1 = dm

dn + 1 = dm

1 = dm - dn

1 = d ( m -n)

d = db, b (m-b) element of Z

=> d|1

=> d <= 1

but d >= 1. so, d = 1.

Is this Correct? pls correct me if im wrong.. do i miss something ?