An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$

$$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$

Clearly the RHS is an (uncountable) infinite sum of closed intervals. I have no idea how to show it is open at two ends.

(My hope is that if this is true, then it is trivial that:

$$f(x)\in\mathscr C^r(a,b)\Longleftrightarrow f(x)\in\mathscr C^r[a+\delta,b-\delta]\quad \forall \delta\in(0,\frac{b-a}2) $$

which will be very useful for me.

A proof that is neat (without too much set theoretical jargon) and based on the original definition will be sincerely appreciated. Thanks in advance!

EDIT. Thanks for Surb’s answer. Now I understand why LHS and RHS are equal. But I’m still confused about another question:

Since they two are equal, doesn’t it imply that if a statement A is true on LHS then it is also true on RHS? However, there are many counter-examples. Like uniform continuity for a function, or uniform convergence for a functional sequence on RHS doesn’t imply that on LHS, but… How come?

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