in courant's and fritz's calculus text im given the assignment to show the above, but first in the same question im given this task: 1) a) if a is rational and x is irrational then x+a is irrational and if a isnt 0 then ax is irrational too. well this task is ofcourse trivial. i thought that these two tasks are connected, although ive seen other proof which do not use taks a. even so, here's the proof of mine which does incorporate a. let's assume a is rational and x is irrational. 1) if a>x>0 then a-x>0 and a-x<a. all we need to show is that a-x is irrational and from a) above, we need to show that -x is irrational, assume it's not, then -x=p/q and x=-p/q contradiction to x being irrational. 2) if x>a>0 then 1>a/x>0 again we need to show that 1/x is irrational, suppose it's not then 1/x=p/q x=q/p and thus a contradiction to x being irrational. is this proof valid? btw, i know other proofs which are quite different than this, so i wonder if this one is valid also.