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.

Ok, let us say you know
$a-\delta_2 < a-\delta <x <a $
Then,
obviously
$a-\delta+2 < a -\delta <x$
Now, to complete it, we keep in mind the transitive property of comparison.
So
\begin{align}
a-\delta+2 < a -\delta &<x\\
&<a\\
&<a+\delta &\text{since $\delta>0$}\\
\end{align}
Now, $\delta = \min(\delta_1, \delta_2) \leq \delta_1$.
So
\begin{align}
a+\delta \leq a+\delta_1
\end{align}
If you know the second case, just do it in reverse.
