Verified answer

Every x ∈ Q can be written as x = r / n where r and n are integers and n ≠ 0.

Since r / n = (-r) / (-n), it is OK to assume that n is a positive integer. That is to

say, we can assume that n ∈ L and r ∈ Z.

Then x = r / n ∈ M_n. So Q ⊂ ∪_(n ∈ L) M_n

Clearly each M_n ⊂ Q, so ∪_(n ∈ L) M_n ⊂ Q.

It follows that Q = ∪_(n ∈ L) M_n

Let x ∈ M_n. Then x = r / n for some r ∈ Z.

So x = (r m) / (m n) for any integer m > 0.

Since r m ∈ Z and m n is a positive integer,

x ∈ M_mn

So M_n ⊂ M_mn

Similarly M_m ⊂ M_mn.