How to use the Lemma of Heine–Borel in order to

Kassandra Ross

Kassandra Ross

Answered question

2022-06-13

How to use the Lemma of Heine–Borel in order to prove the existence of an irrational number?

Answer & Explanation

grcalia1

grcalia1

Beginner2022-06-14Added 23 answers

Here is one possibility. Suppose all the numbers in [0,1] are rational. We know the rational numbers in [ 0 , 1 ] are countable, i.e. they can be enumerated { q 1 , q 2 , q 3 , }. Enclose q 1 in an open interval of width 1 / 2, q 2 in an open interval of width 1 / 4, q 3 in an open interval of width 1 / 8, etc. Since we are assuming all the numbers in [ 0 , 1 ] are rational, this is an open cover of [ 0 , 1 ], hence there is a finite subcover. The subcover has a total width strictly less than 1 / 2 + 1 / 4 + 1 / 8 + = 1. But you can't cover [ 0 , 1 ] with a finite set of intervals of total width less than 1. Contradiction.

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