Cell structure of join spheres

Question

I'm trying to show that the join $S^n * S^m$ is $S^{n+m+1}$ via seeing them as CW complexes and checking that they have the same cell structure. I'm aware of other proof of this homeomorphism but I want to use cell structures. In particular, following Hatcher's book a cell structure of $S^n * S^m$ is: $S^n$, $S^m$ are subcomplexes and the rest of cells are the ones of $S^n \times S^m \times (0,1)$. I know that in a natural cell decomposition of the product, the cells of $X\times Y$ are the cells $e_X \times e_Y$, with $e_X$ a cell of $X$ and $e_Y$ of $Y$, but in this case, what is a cell structure of $(0,1)$? I thought that as it is homeomorphic to $\mathbf{R}$ and this one has a cell structure consisting of the integers as 0-cells and then the intervals between them as 1-cells the same holds for $(0,1)$ via the inverse homeomorphism taking it to $\mathbf{R}$, but then, solving the above problem seems very hard.