Prove that between every rational number and every irrational number there is an irrational number. I have gotten this far, but I'm not sure how to make it apply to all rational and irrational numbers....
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BTW, I'm quite newbish so please explain your reasoning to me like I'm 5. Thanks! 
UPDATE:


Let $p/q$ be a rational number and $r$ be an irrational number.
Consider the number $w = \dfrac{p/q+r}2$ and prove the following statements.
$1$. If $p/q < r$, then $w \in ]p/q,r[$. (Why?)
$2$. Similarly, if $r < p/q$, then $w \in ]r,p/q[$. (Why?)
$3$. $w$ is irrational. (Why?)
