Here, rewrite the rational number as p/q with p and q both integers, q nonzero. If the irrational number is z then their sum is
x = p/q + z
Suppose x is rational, rewrite as a/b like above
a/b = p/q + z
a/b - p/q = z
(aq - pb)/bq = z
We have just shown that z is rational. This is a contradiction. QED.