Let $r > 0$. Then,
(1) The $r$-neighborhood of a set $A$ is the set $N_r(A)$ consisting of all $q$ such that $d(p,q)<r$.
Is there a similar definition for a $r$-interior? Such as
(2?) The $r$-interior of a set $A$ is a set $I_r(A)$ consisting of all $q$ such that $d(p,q)<r$ implies $p \in A$.
That is, $I_r(A)$ contains all points such that an $r$ open ball is contained in $A$.
The main use of the definition is for this theorem:
(3?) If the is a $r>0$ such that $x_n$ is in the $r$-interior of a set $A$, then $d(x_n, y_n) \to 0$ implies $y_n$ is in $A$ for $n$ large enough.
Proof: We have a $N$ such that for $n > N$, $d(x_n , y_n) < r$, and therefore $y_n \in B$ for $n > N$.
The phrase "for some r > 0" is awkward and confusing.
Do you mean that for all r > 0, the r-interior of A,
I(A) = { x : B(x,r} subset A }?
No, I have not seen any use of it. The r-nhood is used.
If A is bounded and r large, then I(A) is empty.