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

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grcalia1 · Accepted Answer

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&#039;t cover   [  0  ,  1  ] with a finite set of intervals of total width less than   1. Contradiction.

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

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