Let $e^{2 \pi i k s} = f(s)$, $f \colon [0,1] \to S^1$ subset of Complex numbers ($S^1$ = unit circle at origin). So $f$ is a loop at the basepoint $1$ in $S^1$. Show that it is not homotopic to the constant loop $c(s) = 1$. This is obvious, but what is a more rigorous way of showing it. Oh yeah, $k$ is an an integer not equal to $0$.
Without reference to complex analysis (I'm looking for a really simple kid proof).