Let $f:I\to \mathbb{R}^2 $ be a continuously differentiable simple closed curve in the plane with a nowhere-vanishing tangent vector (that is, $f(0) = f(1), f'(0) = f'(1)$, and $f'(t) \neq 0 $.) Let $w:I \to \mathbb{S}^1$ be the closed path defined by $w(t)=f'(t)/||f'(t)||$ Prove that $[w]$ is a generator of $\pi_1(\mathbb{S}^1)$.
My idea is to show that $w$ is homotopic to the map $t \to (cos2\pi t),sin(2\pi t)).$ Also I think we need to use mean value theorem . But unable to solve.