In the construction of the lifting at each step it is assumed that the new segment is homeomorphic to some neighborhood of the last point where the path is already constructed, so that the unique path in that neighborhood can be uniquely lifted up, however, if there is no such neighborhood for a point, then the construction may be not possible. For that particular example if we take $e_{0}=1$ and the path $f(t)=(\cos2\pi t,-\sin2\pi t)$ , then however we split the unit interval there must be a subinterval $[s_{i},s_{i+1}]\ni1/2$ , and as it is easy to see the lifting $\tilde{f}$ of $f$ up to this point should be $\tilde{f}(t)=1-t$ , so that $\tilde{f}([s_{i},1])$ should be a connected interval of the form $[k;k+1-s_{i}]\subset(0,+\infty)$ , containing $1-s_{i}$ , which is not possible.