Proof about real numbers

Content

Just a recording of the proof on following topic: between any two distinct real numbers there is an irrational number.

Proof. Let $a,b\in \mathbb{R}$ with $a<b$, and we could pick up a midpoint $q=\frac{a+b}{2}$, which is also in $\mathbb{R}$.
Further, let a small enough irrational number $ϵ=\frac{b-a}{4\sqrt{2}}$, as we know $q+ϵ$ is an irrational number, we have to prove that $q+ϵ<b$.

Just prove that $b$ is larger than $q+ϵ$,

Easily, we know that $4-\sqrt{2}>0$ and $b-a>0$ by definition (two distinct real numbers, thus $a<q+ϵ<b$, meaning that there is an irrational number between two distinct real number.

References

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