Every vector bundle is globally trivial

Cover $S^1$ by the two standard arcs $U_0=S^1\setminus \{(-1,0)\}$ and $U_1=S^1\setminus \{(1,0)\}$. Their overlap has an upper and a lower component. Define a rank-one cocycle by $g_{10}=1$ on the upper overlap and $g_{10}=-1$ on the lower overlap. By [L1], this glues a smooth line bundle $L\to S^1$.

If $L$ were trivial, then [L2] would give a nowhere-zero global frame. In local trivializations that would be given by nowhere-zero functions $f_0$ on $U_0$ and $f_1$ on $U_1$ with $f_1=f_0$ on the upper overlap and $f_1=-f_0$ on the lower overlap. Since $U_1$ is connected, a nowhere-zero continuous $f_1$ has constant sign, but the two overlap equations force opposite signs. This contradiction shows that $L$ is not trivial.