A segment is drawn through each point of the interval $(0,1)$. Prove that the sum of the lengths of these segments is infinite.
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Segments may overlap. Try that. – Mikael Helin Jul 27 '20 at 10:05
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$\sum_{x \in (0,1)} l(x)=\infty$ if $l(x) >0$ for all $x$. In fact this is true if $l(x) >0$ for uncountably many $x$. – Kavi Rama Murthy Jul 27 '20 at 10:06
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Consider the initial segment to be the interval $(0,1)$. The length of sums of this inital segment is 1. Now split the segment in half, the sum of the lengths of the split segments is one. Repeat splitting into halves, after each split the sum of the lengths of the segments are 1. You make infinite number of splits and summing to 1, so you add an infinite number of ones to obtain infinity, i.e. $\sum_{n=0}^\infty 1=\infty$.
Mikael Helin
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