Local degrees for $n=1$.

Question

In Hatcher, p. 136, prop. 2.30, the author needs the fact that $id_{S^n}$ induces an isomorphism $H_n(S^n) \to H_n(S^n,S^n \setminus \{x\})$. For $n>1$ this map sits in the following part of the exact sequence in homology: $$0 = H_n(S^n \setminus \{x\}) \to H_n(S^n) \to H_n(S^n, S^n \setminus \{x\}) \to H_{n-1}(S^n \setminus \{x\})=0$$ which is the result. At $n=1$ however we are left with $$0 = H_1(S^1 \setminus \{x\}) \to H_1(S^1) \to H_1(S^1, S^1 \setminus \{x\}) \xrightarrow{\varphi} H_0(S^1 \setminus \{x\})=\mathbb{Z} \to \mathbb{Z} \to 0 \to 0$$ and since surjective maps $\mathbb{Z} \to \mathbb{Z}$ are injective this gives $\varphi =0$ and the result follows, but Hatcher doesn't mention this last little argument, which makes me wonder if I'm missing something.