Prove that for every irrational number x and rational number

Prove that, for every irrational number x and rational number y such that y y notequalto 0, the product x middot y is irrational.

Solution

Let, x.y be rational

Given y is rational, hence, 1/y is rational

So, (x.y).(1/y) =x is rational as product of rationals is rational

But x is irrational

HEnce contradictoin

So, x.y is irrational