Okay so here are a few definitions first.
Definitions: If $X$ is a nonempty topological spaces, a cell decomposition of $X$ is a partition $\Gamma$ of $X$ into subspaces that are open cells of various dimensions, such that for each cell $e \in \Gamma$ of dimension $n \geq 1$, there exists a continuous map $\Phi$ from some closed $n$-cell $D$ into $X$ (called a characteristic map for $e$) that restricts to a homeomorphism from $\operatorname{Int}D$ onto $e$ and maps $\partial D$ into the union of all cells of dimensions strictly less than $n$. An open cell $e \in \Gamma$ is called a regular cell if it admits a characteristic map that is a homeomorphism onto $\bar{e}$.
Now the way that a regular-cell decomposition of $\mathbb{S}^n$ is usually constructed is by induction. We show that $\mathbb{S}^0$ is a regular $0$-dimensionsl cell complex with two $0$-cells, $\{-1\}$ and $\{1\}$ and then suppose by induction that we can construct a regular cell-decomposition of $\mathbb{S}^{n-1}$ with two cells in each dimension $0, \dots, n-1$.
Then we attempt to construct a regular-cell decomposition of $\mathbb{S}^n$ utilizing our induction hypothesis. To do this we note that we can partition $\mathbb{S}^n$ into the upper hemisphere $H = \{x \in \mathbb{S}^n \ | x_{n+1} > 0\}$, the lower-hemisphere $L = \{x \in \mathbb{S}^n \ | x_{n+1} < 0\}$ and the equator $E = \{x \in \mathbb{S}^n \ | x_{n+1} = 0\}$. Then we note that $H \cong \mathbb{B}^n \cong L$ and $E \cong \mathbb{S}^{n-1} = \partial(\overline{\mathbb{B}^n})$.
Thus $H$ and $L$ are open $n$-cells. In order to show that $H, L$ are regular $n$-cells, I need to provide characteristic maps for each that are homeomorphisms onto their closures. That is I need to find closed $n$-cells $D_1$ and $D_2$ and maps $f_1 : D_1 \to \overline{H}$, $f_2 : D_2 \to \overline{L}$ such that $f_1$ and $f_2$ are homeomorphisms and such that $f_1$ maps $\partial D_1$ into the union of all cells of dimensions stricly less than $n$ (similarly for $f_2$).
Now since $\overline{H} = H \cup E \cong \overline{\mathbb{B}^n}$ and similarly $\overline{L} = L \cup E \cong \overline{\mathbb{B}^n}$, letting $f_1 : \overline{\mathbb{B}^n} \to H$ and $f_2 : \overline{\mathbb{B}^n} \to L$ be the homeomorphisms above, I could claim that $f_1$ and $f_2$ are the required characteristic maps for $H$ and $L$ respectively. However there is a problem.
How do we know that $f_1$ maps $\partial(\overline{\mathbb{B}^n})$ into the union of all cells of dimensions less than $n$, when we have no idea what our (regular) cell-decomposition actually looks like? (Because of our induction hypothesis)
We know that $f_1\left[\partial(\overline{\mathbb{B}^n})\right] = f_1[\mathbb{S}^{n-1}] = E$. So we can simplify the question above to be: Does $E$ contain all the cells of dimensions strictly less than $n$? If so how do we know that?