I'm reading the book Algebraic Topology from a Homotopical Viewpoint but I don't understand why the $ n \in \mathbb{Z} $ in Proposition 2.4.4 is unique. It suffices to prove that $ \widehat{\phi}_n $ and $ \widehat{\phi}_m $ are not homotopic where $$ \widehat{\phi}_k : \mathbb{S}^1 \rightarrow \mathbb{S}^1\ \qquad \widehat{\phi}_n (e^{i2\pi t}) = e^{i2k\pi t} $$ for $ n \neq m $. Is this obvious ?
