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#### OhMyMarkov

##### Member

- Mar 5, 2012

- 83

I want to prove that between two reals, there exists an irrational. This is what I got:

$\forall x \in R$, $\exists m \in Z$ s.t. $m-1 < x < m$. In particular, $x\notin Q$.

$\exists a,b \in R$ s.t. $ax$ and $ax+b$ are irrational. Also, $a(m-1)<ax<am$, $a(m-1)+b<ax+b<am+b$.

End of proof.

I also want to prove that between two reals, there are infinitely many rationals and irrationals. This is my proof:

(Using (1) the above theorem, and (2) the theorem that says that there exists a rational between every two reals)

Combining (1) and (2), between two reals, there exists a real. $\forall x,z \in R$ where $x<z$, $\exists y \in R$ s.t. $x<y<z$. In other words, $\forall x_0, x_1 \in R$ where $x_0 < x_1$, $\exists x_2 \in R$ s.t. $x_0<x_1<x_2$. Apply this again, $x_1<x_2<x_3$. Apply this N-times, $x_N<x_{n+1}<x_{N+2}$. If we let N grow indefinitely, we can say that there are infinitely many reals between two reals.

Are these proofs correct? I am very new to establishing proofs in analysis. Any comments or criticism is highly appreciated.