So now you have two real numbers x and y, and you want to show that Notice first that if r, s are rational numbers with r<x and s<y then r+s<x+y. You also know that (because the index laws are known to hold for rational powers). Taking the sup over all such r and s, you deduce that .
To get the reverse inequality, you need to show that if t is any rational number less than x+y, then it is possible to express t in the form t=r+s, with r<x and s<y. It then follows that . Taking the sup over t, you get
Notice that if t < x+y then t – y < x. Therefore there is a rational number r with If you define then s<y. That justifies the assertion at the start of the previous paragraph (about expressing t in the form r+s).