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An infinite number of points can definitely cover a finite length – any line segment satisfies that condition.

An infinite number of collinear connected (closed) line segments can definitely cover a finite length – one can always subdivide any segment infinitely many times.

But can an infinite number of collinear disconnected (open) line segments cover a finite length?

For completeness, I'm using the Wikipedia definitions for open segment and closed segment, and by "cover a finite length" I mean that all points in the (infinitely many) segments should belong to a finite line segment.

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You can consider $\bigcup\limits_{n=1}^\infty\left(\frac1{n+1},\frac1n\right)\subseteq (0,1)$.

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