I'm trying to see if this is true, but not sure how to go about it. I know that $f$ induces a map on chains where $C\_1(S^1) \to C\_1(S^1)$ is given by $\sigma \mapsto f\circ \sigma$.

How would I go about showing that $f\_\*: H\_1(S^1) \to H\_1(S^1)$ is given by multiplication by $n$?