I have to prove the Borsuk-Ulam theorem following some specific steps. The theorem says that:
For every continuous function $f: S^n \rightarrow \mathbb{R}^n$ there exists $x \in S^n$ with $f(-x)=f(x)$.
I am using the cover of $\mathbb{R}P^n$ by $S^n$. I showed every singular simplex $\tilde\sigma$ of $\mathbb{R}P^n$ can be lifted to $\sigma$ and $\tau \sigma$ singular simplexes of $S^n$.
This induces an application $t:C_{\star}(\mathbb{R}P^n,\mathbb{F}_2)\rightarrow C_{\star}(S^n,\mathbb{F}_2), \tilde\sigma \rightarrow \sigma + \tau\sigma$ which is a chain complex morphism and induces a short exact sequence $$0 \rightarrow C_{\star}(\mathbb{R}P^n,\mathbb{F}_2)\rightarrow C_{\star}(S^n,\mathbb{F}_2) \rightarrow C_{\star}(\mathbb{R}P^n,\mathbb{F}_2) \rightarrow 0$$ where the latter morphism is induced by the covering map. I managed to show all this. This implies directly existence of a long exact sequence in homology.
I should then show that $t \circ p_\star: C_{\star}(S^n,\mathbb{F}_2) \rightarrow C_{\star}(S^n,\mathbb{F}_2)$ is zero where $p_\star$ is induced by the covering map. Can you give me a hint for this?
I should then show that in the long exact sequence in hompology $$0=H_{n+1}(\mathbb{R}P^n,\mathbb{F}_2)\rightarrow H_{n}(\mathbb{R}P^n,\mathbb{F}_2) \rightarrow H_{n}(S^n,\mathbb{F}_2)\rightarrow H_{n}(\mathbb{R}P^n,\mathbb{F}_2)\rightarrow H_{n-1}(\mathbb{R}P^n,\mathbb{F}_2)\rightarrow\cdots$$ the first and third morphisms are zero and the second and fourth isomorphisms. As the second is induced by $t$ and the third by $p_\star$, I could already show that $p_\star$ is zero, as $t$ is injective and the composition $t \circ p_\star$ zero. Thus $t$ is an isomorphism. Furthermore, the fourth morphism, $\delta$ is injective. But how do I show its surjectivity?
Finally I should conclude that if $f: S^n \rightarrow S^m$ is continuous such that, for every $x \in S^n$, $f(-x)=-f(x)$, then $n\leq m$, using "naturality of the long exact sequence". Could you please give me some hints?