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$\begingroup$ When you say "this is nonsense" you need explanation. One explanation is existence of Lebesgue measure. What you show is: a countable set has measure at most $1$. You also need: the real line has measure ${}> 1$. But (so far) you have not proved that. $\endgroup$

Gerald Edgar
– Gerald Edgar

2017-10-06 11:54:44 +00:00
Commented Oct 6, 2017 at 11:54
2

$\begingroup$ I agree there's a detail there, but I think it's straightforward. Measure theory makes it immediate, but isn't necessary. For example, if you use open intervals, you could take a finite subcover of [0,2] by compactness. Now you have a finite set of intervals with total length < 1, which cover an interval of length 2. $\endgroup$

Russ Woodroofe
– Russ Woodroofe

2017-10-06 12:38:03 +00:00
Commented Oct 6, 2017 at 12:38

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