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Let $a<\delta$ and $-\delta <b$ for some $a,b\in \mathbb{R}$ and $\delta >0$. Is there a $\lambda \in [0,1]$ so that $-\delta \leq \lambda a +(1-\lambda )b\leq \delta$?

M.Ramana
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  • @Surb How do you conclude that $[a,b] \subset (-\lambda,\lambda)$? – mathcounterexamples.net Mar 11 '22 at 13:01
  • @Surb Generally, $a$ is not less than $b$. – M.Ramana Mar 11 '22 at 13:04
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1 Answers1

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If $b\le a$ then $-\delta < b \le a < \delta$, so $\lambda=0$ will do. If $b>a$ then $c=\max(a,-\delta) \in [a,b] \cap [-\delta,\delta]$ and can be written as a convex combination of $a$ and $b$.

Jamie Radcliffe
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  • This question seems not to meet the standards for the site. Instead of answering it, it would be better to look for a good duplicate target, or help the user by posting comments suggesting improvements. Please also read the meta announcement regarding quality standards. – Shaun Mar 11 '22 at 13:27