I'm wondering if I went wrong in this proof, and if so where. I consider a continuous map $f: S^n \to \mathbb{R}^n$, and by continuity the set $f(S^n)$ is compact and connected in $\mathbb{R}^n$.
Now assume for now that this set is in fact homotopy-equivalent to $B_1(0)$, the closed unit ball in $\mathbb{R}^n$. Then consider the map which sends $f(x) \in f(S^n)$ to $f(-x) \in f(S^n)$. This map is continuous and maps the aforementioned compact set (homotopy-equivalent to a closed ball) to itself. Then Brouwer's theorem applies, guaranteeing a fixed point of the map, i.e. f(x) = f(-x).
Then I justify the claim of homotopy equivalence by saying that the map $f: S^n \to \mathbb{R}^n$ induces a homomorphism between the homology groups of the respective spaces, and $S^n$ only has nontrivial $H^n$ (but $\mathbb{R}^n$ has trivial $H^n$) so we have the aforementioned compact sets having all homology groups trivial (so the set is contractible).
This proof seems too sketchy to work, so please criticize all the errors you see.