Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 2F / b / i / Solution

Put $z(t)=e^{2\pi it}$ and use the straight homotopy $H(s,t)=z(t)+s(\gamma (t)-z(t))$. For $0\le s<1$, the triangle inequality givesAt $s=1$ it is nonzero by the hypothesis. Thus this is a homotopy of closed loops avoiding the origin, even though the given distance bound is non-strict. The homotopy invariance of winding number gives