Let x=(m1/m2)+p and y=(n1/n2)+q, where m1, m2, n1, n2 are integers and m2,n2≠0. The fractions are rational by definition of rational numbers; p and q are irrational so cannot be expressed as the quotient of two integers.
x+y=(m1/m2)+(n1/n2)+p+q=(m1n2+m2n1/m2n2)+p+q=M/N+p+q where M is an integer = m1n2+m2n1 and N is also an integer = m2n2. Therefore M/N is rational and p+q is irrational. So x+y is irrational. However x and y are individually irrational; neither is rational.
Let p=0 so x is rational. In this case x+y=M/N+q which is irrational because of q. The sum is irrational because y is irrational. Similarly, if q=0, x+y is irrational because x is irrational.
If p=q=0 then x+y is rational.
All cases of rationality have been covered. If x+y is irrational, therefore, either x or y are irrational, or both are irrational, so it is not true that at least one is rational. Rather, it is true that at least one is irrational.