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We have the two real intervals [a,b] and [c,d] as shown below,

enter image description here

From figure it is very clear that $c\leq a$ and $b \leq d$.

I wanted to know what one can give argument on it why is $c \leq a$ and $b \leq d$?

1 Answers1

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First note $I=[a,b]\subset J=[c,d]$. Suppse $a<c$, then $\exists e\in \mathbb{R}$ such that $a<e<c$. Then $e\in I, e\not\in J$ by definition of interval, which contradicts to $I\subset J$. Hence $a\geq c$. Similarly for $b,d$.

John
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  • you have stated that I $\subset$ J, can you argu why I $\subset$ J? – Willim Smith Nov 18 '14 at 16:46
  • @WillimSmith From the picture, we know $c\leq a, b\leq d$, then use the definition of interval, $[a,b]=\{x| a\leq x\leq b\}$ – John Nov 18 '14 at 16:58
  • I mean why I $\subset$ J, you used the definition of interval can you please explain it a bit more keeping in mind why I $\subset$ J? How one can prove it that I $\subset$ J? – Willim Smith Nov 18 '14 at 18:03