# Proof Irrational number x , given N &#x2208;<!-- ∈ --> <mrow class="MJX-TeXAtom-ORD">

antennense 2022-07-11 Answered
Proof Irrational number $x$, given $N\in \mathbb{N}$ exists $ϵ>0$ so all rationals $\in {V}_{ϵ}\left(x\right)$ have denominator larger than $N$.
You can still ask an expert for help

## Want to know more about Irrational numbers?

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it

## Answers (1)

Jenna Farmer
Answered 2022-07-12 Author has 17 answers
Since there are only finitely many rational points in $\left[0,1\right]$ whose denominator is not larger than $N$ there has to be a $\epsilon$ so small that $\left(x-\epsilon ,x+\epsilon \right)$ contains none of them; just take

We have step-by-step solutions for your answer!

Expert Community at Your Service

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Available 24/7
• Math expert for every subject
• Pay only if we can solve it