I am reading Bertsekas' Convex Analysis and Optimization. The book requires some basic knowledge of real analysis such as norm equivalence property in real coordinate space and outer limit/ inner limit of sequences of sets. Though there are references to Ash's Real analysis and Probability and Rudin's principle of mathematical analysis, these knowledge seems not defined in the two books. I have searched the functional analysis textbook by Kreyszig and found the equivalence of norm. Nevertheless the outer limit/ inner limit are not explored ( Ash's book use another definition.) Are there other textbooks developed these ideas more thoroughly?
Let $\{X_n\}$ be a sequence of nonempty subsets of $\mathbb{R}^n$. The outer limit of $\{X_n\}$ is the set of all $x \in \mathbb{R}^n$ such that every neighborhood of $x$ has a nonempty intersection with infinitely many of the sets $X_k$, $k=1,2,3 \cdots $. Equivalently, outer limit is the set of all limit points of sequence $\{x_k\}$ such that $x_k\in X_k$, $k=1,2,3 \cdots $
The inner limit of $\{X_n\}$, is the set of all $x \in \mathbb{R}^n$ such that every neighborhood of $x$ has nonempty intersection with all except finitely many of sets $\{X_n\}$, , $k=1,2,3 \cdots $. Equivalently, inner limit is the set of all limits of convergent sequence $\{x_k\}$ such that $x_k\in X_k$, $k=1,2,3 \cdots $
