Discrete math contradiction question.?

say r,s are in R , r is irrational and s is rational.

Now assume (in spirit of finding a contradiction), that r/s is rational. If r/s is rational, then there exist p,q in Z such that

r/s = p/q

This means, however, that r = sp/q.

Because s is rational, s = a/b, thus r = ap/qb

This means however, because a,b,p,q are in Z, that r is a rational number. This is a contradiction, because we've assumed that r was irrational.

So r/s is irrational

Extra condition: s <> 0;

Assume the statement is false, hence r/s is a rational number q.

r=qs which is rational since the product of two rational numbers is a rational number (why?)

This is a contradiction with the hypothesis which states that r is irrational.

Thus, the statement is true.