Prove the following statement by contradiction For all real

Prove the following statement by contradiction:

For all real numbers r and s, if r is rational and s is irrational, then r+2s is irrational.

Solution

suppose that r+2s is rational, where r is rational and s is irrational then as we know that sum of rationals is rational, so

r+2s-r=2s is rational and 2s/2=s is rational, which is wrong as s is given irrational to us, so our supposition is wrong that r+2s is rational

=> r+2s irrational