How to understand out of bound in the following theoretical context?
It is often useful to allow arrays of zero length. These are just empty arrays. The length of the subarray $A[i\ldots j]$ is $j-i+1$, so when $j=i-1$, we just get an array of length 0.

Why is this useful? Here are some examples:


An array $A[1\ldots n]$ can always be partitioned into two arrays $A[1\ldots i]$ and $A[i+1 \ldots n]$ of lengths $i,n-i$. This works even for $i \in \{0,n\}$.
We can construct an array $A[1\ldots n]$ inductively using the formula $A[1\ldots i] = A[1\ldots (i-1)] \cdot A[i]$. The base case is the empty array $A[1\ldots 0]$.


Having arrays of negative length makes less sense, and could lead to confusion and errors, since there is no semantics under which the formula $|A[i\ldots j]| = j-i+1$ would hold for them, since array lengths are non-negative.