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

antennense

antennense

Answered question

2022-07-11

Proof Irrational number x, given N N exists ϵ > 0 so all rationals V ϵ ( x ) have denominator larger than N.

Answer & Explanation

Jenna Farmer

Jenna Farmer

Beginner2022-07-12Added 17 answers

Since there are only finitely many rational points in [ 0 , 1 ] whose denominator is not larger than N there has to be a ε so small that ( x ε , x + ε ) contains none of them; just take
ε = min { | x y | : y  is rational and its denominator is N } .

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