My question pertains to the paper ["A Simplified Proof of the Divergence Theorem"](http://docdro.id/E0I6BtA) by Djairo Guedes de Figueiredo.


It's not a big question, actually, but it's confusing me a lot: In the statement of Lemma 2 in the paper, the author writes "$h(x') < x\_1 < 1$ and $-1 < x' < 1$", where $x'=(x\_2,x\_3,\ldots,x\_n)$. 


Why the statement $-1 < x' < 1$ makes sense? Because we can't compare vectors with numbers using inequalities: $-1$ and $1$ are numbers and $x'$ is a coordinate pair of the last $n-1$ coordinates of the vector $x=(x\_1,x')=(x\_1,x\_2,\ldots,x\_n) \in \mathbb R^n$.



Comparing vectors with numbers?
The statement $-1 < x' < 1$ means that every coordinate of $x'$ is between $-1$ and $1$. (The set of all such $x'$ is a cube.)


