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In Stillwell's Mathematics and Its History we read

The need for clarification arose from the discovery of Harnack (1885) that any countable subset {x0, x1, x2, . . .} of R could be covered by a collection of intervals of arbitrarily small total length. Namely, cover x0 by an interval of length ε/2, x1 by an interval of length ε/4, x2 by an interval of length ε/8, . . . , so that the total length of intervals used is ≤ ε. (This is another proof, by the way, that R is not a countable set.)

I could not find a reference expounding such a proof, can anybody point to one?

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  • $\begingroup$ Does this method actually prove that $\mathbb R$ is uncountable ? And if yes, how can we conclude that with this argument ? $\endgroup$
    – Peter
    Commented Jan 24, 2018 at 14:36
  • $\begingroup$ If $\mathbb{R}$ were countable, it could be covered by a set of arbitrarily small total length, which is absurd. $\endgroup$
    – Hans Engler
    Commented Jan 24, 2018 at 15:01
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    $\begingroup$ Why is it absurd? The Cantor set is uncountable, and can be covered by sets of arbitrarily small total length. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 24, 2018 at 15:05
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    $\begingroup$ I don't see why you need a reference; you've given more or less the entire proof. The only missing detail is showing that $\Bbb R$ cannot be covered by intervals of small total length; that follows easily from compactness of $[0,1]$. $\endgroup$
    – David C. Ullrich
    Commented Jan 24, 2018 at 15:09
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    $\begingroup$ Oxtoby's little book has the missing argument (if $[0,1] $ is covered by open intervals, the sums of their lengths is at least 1). The argument does not requore uncountability of $\mathbb R $. $\endgroup$
    – Andrés E. Caicedo
    Commented Jan 24, 2018 at 15:15

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