Solution
= 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> gives
$$
|H(s,t)|\ge1-s|\gamma(t)-z(t)|\ge1-s>0.
$$
At $s=1$ 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
$$
\boxed{w(\gamma;0)=w(e^{2\pi it};0)=1.}
$$