Saturday, 1 April 2017

real analysis - Proof on density of rationals and irrationals in $mathbb{R}$ via dedekind cut



Assume $a$ is a dedekind cut. By definition it is nonempty so there exist $b\in a$. Similarly, we can find $p\in a$ such that $p>b$. It has no maximals element so, we can repeat this process to infinity.



Claim: There exist atleast one irrational between two rationals.




Consider $p$ and $b$ that we talked about earlier. W.l.o.g say both are positive and observe that $1>\frac{1}{\sqrt2}>0$ so we can find $n\in \mathbb{N}$ such that $p-b>\frac{1}{\sqrt2 n}>0$, by archimedean property. Now by adding b to the inequality, we get $p>\frac{1}{\sqrt2 n}+b>b$. (It is trivial that $\frac{1}{\sqrt2 n}+b$ is irrational).



As we can find infinitely many rationals that are greater than other rational in our dedekind cut $a$(because dedekind cut has no maximal element) we can find infinitely many irrationals under a cut, in our case $p$,$b$$

So my question is that, is this proof valid? if not, then why?


Answer



$1 > \sqrt(2) > 0$
is not true, is it a mistype?



Anyway, it could be a lot simpler. Consider this number:




$a + \frac{(b - a)}{\sqrt(2)}$



This gives you an algebraic irrational between a and b. Use $\pi$ in place of $\sqrt(2)$ if you want a transcendental.


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