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How can I calculate the supremum of the infimum of a function depending on two variables?

For example:

$$ \begin{align*} a =\sup_{t \geq 0} \left\{ \inf_{r \geq 0} \{ 10 \leq t + r \} \right\} \end{align*} $$

What I would have done is calculating the infimum first, only depending on r and then the supremum only depending on t.

This would give me $a = 10$. Is that correct?

maax
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  • Is $\leq$ a typo here? – Kavi Rama Murthy Apr 23 '19 at 08:39
  • If $10\leq t+r$ is supposed to be $10-t+r$ then your answer is correct. – Kavi Rama Murthy Apr 23 '19 at 08:41
  • The $\leq$ is correct. Would it make sense when defined for natural numbers? – maax Apr 23 '19 at 08:42
  • Does it even make any sense in its current form? – maax Apr 23 '19 at 08:48
  • What's your opinion? The inf of something is roughly the minimum value it can take (I'll leave the details for now, there are other concerns). What is the minimum value of $10\le t+r$? How is $10\le t+r$ even a value? How does it make any sense? – Jean-Claude Arbaut Apr 23 '19 at 08:51
  • @maax The only way I can see of making any sense of the expression $\ \inf_{r \geq 0} { 10 \leq t + r }\ $ is to interpret it as $\ \inf {r,\vert, r\ge0\mbox{ and }10 \leq t + r }\ $, but, in my opinion, it is certainly not a reasonable way of expressing this latter meaning. – lonza leggiera Apr 23 '19 at 09:15

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