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So this is a multipart question, and I have a couple "theories" on how to attack it but still stuck in the infancy stages. Basically, not very far.

For each real $r>0$ let $I_r$ denote the open interval $(0,r)$ and let $J_r$ denote the closed interval $[0,r]$.

  1. Show that the intersection of any finite number of the intervals $I_r$ is nonempty.
  2. Show that the intersection of all intervals $I_r$ is the empty set.
  3. Would your answers to 1 and 2 change if $I_r$ was replaced by $J_r$?

I understand that the intersection of all the intervals $I_r$, there is no number x that belongs to every $r>0$.

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Hint: for $1.$ use induction and for $2.$ contradiction.

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  • $\begingroup$ I see the thinking but am still stuck, im not sure how to apply the intersection as a function... $\endgroup$
    – I am 126264.
    Feb 5, 2014 at 15:26

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