The relative interior is the union of a nested sequence of nonempty compact convex subsets

Question

I read the following statement and its proof on Wikipedia:

If $A \subset \mathbb{R}^n$ is nonempty and convex, then its relative interior $\text{relint}(A)$ is union of nested sequence of nonempty compact convex sets $K_{1} \subset K_{2} \subset \dots \subset \text{relint}(A)$

where $\operatorname{relint}(A) = \{x \in S : B(x, \epsilon) \cap \text{aff}(S) \subset S \}$, $B(x,\epsilon)$ is ball with center at $x$, radius $\epsilon$, $\operatorname{aff}(S)$ is the affine hull of $S$. On Wikipedia, the following proof is provided:

Since we can always go down to the affine span of $A$, WLOG, the realative interior has dimension $n$. Now let $K_{j} = [-j,j]^{n} \cap \{ x \in \text{int}(A) : \operatorname{dist}(x, (\operatorname{int}(K))^{\mathsf{c}}) \geq \frac{1}{j} \}$

I cannot understand at following points:

1. What does "go down to the affine span of $A$" means? Why can we assume that affine dimension of relative interior is $n$?

2. Why $K_{j}$ is convex? It seems hard to use direct definition of convex. How can I prove it?

3. Lastly, why relative interior is union of such $K_j$?