How to argu on the interval of real line?

Question

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

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 $b \leq d$?

Answer 1

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 $b,d$.