How to check if two inequalities express the same thing? Suppose:
$$
\delta_1>0 \\
\delta_2>0 \\
\delta=\text{min}(\delta_1,\delta_2)
$$
If we know that the following is true:
$$
a-\delta_2<a-\delta<x<a \hspace{2cm} \text{ or } \hspace{2cm}  a<x<a+\delta<a+\delta_1
$$
Can we conclude that the followng is true too?
$$
a-\delta_2\leq a-\delta < x < a+\delta \leq a+\delta_1
$$
I don't know how to formally show this.
Thank you in advance for any help provided.

Note that $0<\delta\le \delta_{i}$ for $i=1,2$. So $0>\delta\ge -\delta_{i}$ for $i=1,2$. Thus 
$a-\delta_{2}\le a-\delta$ and $a+\delta\le a+\delta_{1}$. This holds in general by the definitions.
If we know $a-\delta<x<a$ then $a-\delta_{2}\le a-\delta<x<x+\delta<a+\delta\le a+\delta_{1}$.
If we know $a<x<a+\delta$ then $a-\delta_{2}\le a-\delta<a<x<a+\delta\le a+\delta_{1}$.
