Measure of a Cartesian Product of a Null Set and Another Euclidean Space

Measure of a Cartesian Product of a Null Set and Another Euclidean Space

Let $n\in\mathbb{Z}_+$, $n\geq 2$, $k\in\mathbb{Z}_+$, and $k<n$. Let
$N\subset\mathbb{R}^k$ be a $k$-dimensional Borel null set. Consider the
Cartesian product $N\times\mathbb{R}^{n-k}\subset\mathbb{R}^n$.
My intuition suggests that $N\times\mathbb{R}^{n-k}$ must be a null set as
well (using the $n$-dimensional Borel measure, that is), but I cannot seem
to prove it. The problem is that the usual strategy of approximating only
$N$ by ever smaller rectangles from the outside does not work, since
$\mathbb{R}^{n-k}$ has infinite measure (using the $n-k$-dimensional Borel
measure), so that a more sophisticated limit argument must be used, in
which $N$ and $\mathbb{R}^{n-k}$ are jointly approximated by small
rectangles (of the appropriate dimension) to be picked "strategically."
I would appreciate any ideas or hints.